Wolfram Research

Wolfram Research

QuantumFramework

Perform analytic and numeric computation in Quantum Information Theory

Contributed By: Wolfram Research, Quantum Computation Framework team

Quantum Computing is an emerging technology that could revolutionize how computations are performed in the future. It operates based on the law of quantum mechanics - a radically different law of physics from the one our current computers are based on. A quantum computer may be able to solve problems that are difficult to solve using current computers such as breaking encryption, simulating a quantum system, or enhancing machine learning algorithms.

The Wolfram Language Quantum Computation Framework performs high-level analytic and numeric computations in Quantum Information Theory, allowing simulation of quantum circuits and quantum algorithms. Starting from discrete quantum mechanics, this framework can work with quantum states and quantum operators, implements measurements, performs basis manipulation, computes measure entanglement, and more. The semantics is very intuitive and it is equipped with various named states and operators such as Bell states, Pauli operators, universal quantum gates, and others. Thus, making it easy for people to simulate a quantum computer using the Wolfram Language.

Details

This paclet works with V12.3 and more of Woflram Language Mathematica.

Examples

Quantum Basis and State (19) 

In our Quantum Framework, a quantum state is defined with respect to a QuantumBasis. For pure states, the given input can be a list of amplitudes, or an association with keys corresponds to basis elements and values as amplitudes.


Define a 2D quantum basis (computational):

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Out[1]=

Given the basis dimension n, the basis elements will be indexed by ket with i=0,1,...,n-1

In[2]:=
%["ElementAssociation"]
Out[2]=

Use a sequence as QuantumBasis[n,m], for n qudits of m-dimensional (overall dimension will be nm). For example, let us define a 2×2×2 dimensional quantum basis (three qubits):

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Out[3]=
In[4]:=
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Out[4]=

Use a list as QuantumBasis[{n1,n2,...,nm}] for n1×n2×n2×...×nm dimensional Hilbert space of m qudits. For example, define a 3×5 dimensional quantum basis (two qudits):

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Out[5]=
In[6]:=
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Out[6]=

A basis can also be defined by an association with the basis element names as keys, and corresponding vectors as values.

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Out[7]=

There are many named-basis built in the quantum framework, {"Computational", "PauliX", "PauliY", "PauliZ","Fourier", "Identity","Schwinger", "Pauli", "Dirac", "Wigner"}:

In[8]:=
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After a basis object has been defined, it is straightforward to use it to construct quantum states and operators. A quantum state is represented by QuantumState object and a quantum operator is represented by QuantumOperator.

For pure quantum states, a vector with elements as amplitudes, and the corresponding basis should be given in this format: QuantumState[amp_List,QuantumBasis[args]]. With no basis specified, the default basis will be the computational basis whose dimension depends on the amplitude vector.


Define a pure 2-dimensional quantum state (qubit) in the Pauli-X basis:

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Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{1, -1 + I}, InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumBasis\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["PauliX"]]
Out[9]=
In[10]:=
%["Formula"]
Out[10]=

If the basis is not specified, the default is the computational basis of 2n dimension (n qubits):

In[11]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{3, 2 I, 1, -5}]
%["Amplitudes"]
Out[11]=
Out[12]=

If the vector has more than 2 elements, it is interpreted as a n-qubit state (by right-padding of zeros), unless the dimension is specified.

In[13]:=
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{2, 0, I}]
Out[13]=
In[14]:=
state["Amplitudes"]
Out[14]=

Same amplitude vector, but this time the dimension is specified:

In[15]:=
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{2, 0, I}, 3];
state["Amplitudes"]
Out[16]=

Many named states are available for easy access:

In[17]:=
{InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], 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InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], 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  "PsiPlus"], InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["GHZ"]}
Out[17]=

Using associations, one can create a superposition of states, with keys as a list of corresponding indexes, and values as amplitudes. For example, let us create a superposition of 3 qubits (i.e., QuantumBasis[2,3]) as :

In[18]:=
\[Psi] = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][<|{0, 0, 0} -> 1/
    Sqrt[2], {1, 1, 1} -> 1/Sqrt[2]|>, 2, 3]
Out[18]=
In[19]:=
\[Psi]["Formula"]
Out[19]=
In[20]:=
\[Psi]["Dimensions"]
Out[20]=

A different way of creating superposition is simple adding two quantum state objects. For example, the previous state can be constructed as follows, too:

In[21]:=
\[Psi]2 = (InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"BasisState", {0,
         0, 0}}] + InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"BasisState", {1, 1, 1}}])/
   Sqrt[2];
\[Psi] == \[Psi]2
Out[19]=

Once a built-in basis is specified, amplitudes correspond to the basis elements. For example, let's use Bell basis:

In[22]:=
Normal /@ InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumBasis\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["Bell"][
  "ElementAssociation"]
Out[22]=
In[23]:=
\[Psi] = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{2, -I, 1, 3}, InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], 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Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["Bell"]]
Out[23]=
In[24]:=
\[Psi]["Amplitudes"]
Out[24]=

We can also define a state by inputting a density matrix:

In[25]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
 1/2 (IdentityMatrix[2] + PauliMatrix[2])]
Out[25]=

For pure states, one can get the corresponding normalized state vector:

In[26]:=
%["Formula"]
Out[26]=

Define a generic Bloch vector:

In[27]:=
mat[r_] /; VectorQ[r] := 1/2 (IdentityMatrix[2] + r . Table[PauliMatrix[i], {i, 3}])
In[28]:=
r = {.1, .1, 0};
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][mat[r]]
Out[20]=

Test if it is a mixed state:

In[29]:=
state["MixedStateQ"]
Out[29]=

Calculate Von Neumann Entropy:

In[30]:=
state["VonNeumannEntropy"]
Out[30]=

Purity:

In[31]:=
state["Purity"]
Out[31]=

A pure state (by normalizing the vector r):

In[32]:=
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][mat[Normalize@r]]
Out[32]=

Purity:

In[33]:=
state["PureStateQ"]
Out[33]=

Calculate Bloch Spherical Coordinates

In[34]:=
state["BlochSphericalCoordinates"]
Out[34]=

A matrix that is not positive semi-definite (cannot be a density matrix in standard QM, but in ZX-formalism we can have situations like this):

In[35]:=
mat[{1, 2, 0}] // PositiveSemidefiniteMatrixQ
Out[35]=

A non positive-semidefinite matrix as state:

In[36]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][mat[{1, 2, 0}]]
Out[36]=

For the matrices as an input, if no basis is given, the default basis will be computational.

In[37]:=
m = RandomComplex[{-1 - I, 1 + I}, {8, 8}];
\[Rho] = ConjugateTranspose[m] . m;
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][\[Rho]]
Out[34]=
In[38]:=
state["Dimensions"]
Out[38]=

Using ρ, define a quantum state in 2×4 dimensional basis (note the number of qudits):

In[39]:=
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][\[Rho], {2, 4}]
Out[39]=
In[40]:=
state["Dimensions"]
Out[40]=

Define a quantum state in 8D Hilbert space (one qudit, only) :

In[41]:=
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][\[Rho], 8]
Out[41]=
In[42]:=
state["Dimensions"]
Out[42]=

One can also define a state in a given basis, and then transform it into a new basis. For example, let us transform in the computational basis into the basis of Pauli-X {,}:

In[43]:=
\[Psi]1 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{1, 0}];
\[Psi]2 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][\[Psi]1, "PauliX"]
Out[36]=

Return amplitudes:

In[44]:=
\[Psi]1["Formula"]
\[Psi]2["Formula"]
Out[44]=
Out[38]=

Note states are the same, only defined in different basis

In[45]:=
\[Psi]1 == \[Psi]2
Out[45]=

One can use QuantumTensorProduct, to construct different states or operators. Create tensor product of plus state of three qubits :

In[46]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/8a199f0b-b95c-4e9f-b518-0ba85d489303"]
Out[36]=

Another way of defining is to first define a basis, and then assign amplitudes:

In[47]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/1b7939df-573e-457d-9a25-eeef46d3d631"]
In[48]:=
\[Psi]2 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{0, 0, 0, 0, 0, 0, 0, 1}, plusbasis];
\[Psi]2["Formula"]
Out[40]=
In[49]:=
\[Psi]2 == \[Psi]1
Out[49]=

Quantum Operators (11) 

Quantum operators can be define by a matrix, or specifying only eigenvalues with respect to a QuantumBasis. Additionally, there are many built-in named operators.


Define Pauli-x operator:

In[50]:=
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Out[50]=

Action of Pauli-x operator on a symbolic state :

In[51]:=
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Out[51]=
In[52]:=
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Out[52]=

Action of Hadamard operator :

In[53]:=
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Out[53]=
In[54]:=
% == InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["Plus"]
Out[54]=

One can also construct a composition of operators. For example, let's checking the relation :

In[55]:=
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Out[55]=
In[56]:=
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Out[56]=

Multi-qubit operators can take specific orders

Define state α+β:

In[57]:=
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state["Amplitudes"]
Out[56]=

Apply Pauli-X on the 2nd qubit only (by defining order in the operator)

In[58]:=
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Out[58]=
In[59]:=
% == InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][<|{1, 0} -> \[Beta], {1,
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Out[59]=

For multi-qudit cases, one can define order, or construct the operator using QuantumTensorProduct. For example:

In[60]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/446e5797-9cb7-4e70-a53a-4565c3de2311"]
Out[60]=

Generalization of Pauli matrices to higher dimensions:

In[61]:=
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Out[61]=
In[62]:=
%["Matrix"] // MatrixForm
Out[62]=

Generalization of Hadamard to more qubits:

In[63]:=
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Out[63]=
In[64]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/2c9fb727-6207-4e91-bbfd-133460e7e272"]
Out[64]=

Action of Hadamard of 3-qubits on (a very common step in most quantum circuits):

In[65]:=
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Out[65]=
In[66]:=
% == InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"UniformSuperposition",
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Out[66]=

One can define a “ControlledU” operator, with specific target and control qudits.

In[67]:=
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Return control and target qudits:

In[68]:=
AssociationMap[ct, {"ControlOrder", "TargetOrder"}]
Out[68]=

Action of T-controlled (1,2) on :

In[69]:=
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Out[69]=
In[70]:=
%["Formula"]
Out[70]=

Note that "CT" is also a named controlled operator in our framework:

In[71]:=
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Out[71]=

One can create new operator by doing some mathematical operations (e.g,. exponential, fractional power etc) on a quantum operator:

In[72]:=
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Out[72]=

Show that the result is the same as rotation operator around x:

In[73]:=
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Out[73]=

A fractional power of NOT operator:

In[74]:=
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Out[74]=
In[75]:=
%["Matrix"]
Out[75]=

Describing Quantum Measurement (9) 

In our quantum framework, one can study projective measurements or any general measurement (POVM), using QuantumMeasurementOperator.

PVM (projective measurements) (6) 

A measurement can be defined by specifying the corresponding measurement basis.


Measurement of a one-qudit system in the state basis:

In[76]:=
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Out[49]=
In[77]:=
m["ProbabilityPlot"]
Out[77]=

Post-measurement states per measurement result:

In[78]:=
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Out[78]=

Measurement of a two-qudit system in the computational basis:

In[79]:=
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m = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumMeasurementOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumMeasurementOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
   "Computational", {1, 2}][\[Psi]0]
Out[80]=
In[81]:=
m["ProbabilityPlot"]
Out[81]=

One can get the post-measurements states of measurement results:

In[82]:=
m["StateAssociation"]
Out[82]=

For composite systems, one can specify measuring of one or more qudits. It can be done by specifying an order for QuantumMeasurementOperator. For example, measurement of a only 1st qudit of a two-qudit system in the state basis:

In[83]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/3c5ef3a9-9eef-414a-afc1-842cfaa95194"]
Out[84]=
In[85]:=
m["ProbabilityPlot"]
Out[85]=
In[86]:=
m["StateAssociation"]
Out[86]=

Check the post-measurement states are as and

In[87]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/3d07de5b-e9b5-4913-b948-49f09b1a8597"]
Out[87]=

One can use the following format for named basis and specifying corresponding eigenvalues: QuantumMeasurementOperator[name->eigenvalues]. For example, define measurement operator by a QuantumBasis object (as an eigenbasis) and a list of eigenvalues:

In[88]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumMeasurementOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumMeasurementOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
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    I, 2, 3}, 4]
Out[88]=

The representation of quantum state and operator can be changed according to any basis transformation. Basis transformation is applicable to QuantumState, QuantumOperator, and QuantumMeasurementOperator as demonstrated below:

Basis change for a quantum state:

In[89]:=
state = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 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In[90]:=
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In[91]:=
state["Amplitudes"]
Out[91]=
In[92]:=
newState["Amplitudes"]
Out[92]=

Basis change for a quantum operator:

In[93]:=
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In[94]:=
newOperator = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
   operator, InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumBasis\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["PauliX"]];
In[95]:=
operator["Amplitudes"]
Out[95]=
In[96]:=
newOperator["Amplitudes"]
Out[96]=

POVM (3) 

One can also give a list of POVM elements to define measurement operator.

In[97]:=
povm = {o1, o2, o3} = {{{2/3, 0}, {0, 0}}, {{1/6, 1/(2 Sqrt[3])}, {1/(2 Sqrt[3]), 1/2}}, {{1/6, -(1/(2 Sqrt[3]))}, {-(1/(2 Sqrt[3])), 1/2}}};

Check that all POVM elements are explicitly positive semi-definite:

In[98]:=
PositiveSemidefiniteMatrixQ /@ povm
Out[98]=

Check the completeness relations:

In[99]:=
(Plus @@ povm) == IdentityMatrix[2]
Out[99]=
In[100]:=
qmo = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumMeasurementOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumMeasurementOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][povm]
Out[100]=

Post-measurement states after measuring POVMs:

In[101]:=
qmo[InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{1, 1}]]["StatesAssociation"]
Out[101]=

Quantum Circuits (8) 

In quantum information, it is often convenient to have a local state description of a multi-qubit state. For instance, Alice and Bob shared a (possibly entangled) quantum state. But, since they are really far away from each other, they do not have access to each other state. In such scenario, Alice's description of her part of quantum state is described by a "reduced state" that can be obtained by QuantumPartialTrace operation.

Trace out the second subsystem in a two-qubit state:

In[102]:=
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Out[102]=

Partial trace can also be applied to QuantumBasis:

In[103]:=
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Out[103]=

There are several metrics to measure entanglement between two qudits, such as concurrence, entanglement entropy, negativity etc. The calculation of entanglement measure is represented by QuantumEntanglementMonotone function.

Plotting various entanglement measure for the state

In[104]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/3c6d14c6-9375-405f-a9ab-fcc6b760217a"]
Out[104]=

To know whether a state is entangled or separable without computing its measure, QuantumEntangledQ is used.

Checking whether a subsystem 1 and 3 is entangled in "W" state:

In[105]:=
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state["Formula"]
Out[106]=
In[107]:=
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumEntangledQ\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumEntangledQ\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumEntangledQ"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumEntangledQ"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][state, {{1}, {3}}]
Out[107]=

In quantum information, there exist notions of distance between quantum state such as fidelity, trace distance, Bures angle, et cetera. One may compute distance between two quantum state using various metrics by QuantumDistance:

Measuring trace distance between a pure state and a mixed state:

In[108]:=
\[Psi]1 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{{1/4, 0}, {0, 3/4}}];
\[Psi]2 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{1, 0}];
In[109]:=
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Out[109]=

One may create a list of QuantumOperator and/or QuantumMeasurementOperator to build a quantum circuit. A quantum circuit is represented by the symbol QuantumCircuitOperator, that can take single or multi qudit operators.

Example for the construction of controlled quantum gate:

In[110]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/482cd496-2df7-4fe4-83d4-ca5772ecfb22"]
Out[111]=

Decomposition of Toffoli gate:

In[112]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/ad9528f2-5dd7-451f-971d-4536c1caa80b"]
Out[112]=

Show above implementation is the same as Toffoli gate:

In[113]:=
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Out[113]=

One can define more than one control and target qubits.

Define control and target qubits:
In[114]:=
controls = {1, 2, 3};
targets = {5, 6};
Define unitary operator and controlled-u operator:
In[115]:=
u = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["X", targets];
cu = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"ControlledU", u, controls}];
InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumCircuitOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumCircuitOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumCircuitOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumCircuitOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{cu}]["Diagram"]
Out[116]=
Return the control qubits:
In[117]:=
cu["ControlOrder"]
Out[117]=
Return the target qubits:
In[118]:=
cu["TargetOrder"]
Out[118]=

One can add measurement to a quantum circuit, too. For a single qubit unitary operator U with eigenvalues ±1, a measurement of U can be implemented in the following circuit. Note we have considered Pauli-Y as U operator:

In[119]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/85cf79f9-8228-4231-89a5-9ff4b3b6f99a"]
Out[109]=

The result of circuit on a quantum state will be a quantum measurement:

In[120]:=
m = qc[InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"Register", 2}]]
Out[120]=

Let us calculate state of 2nd qubit after the measurement, by tracing over 1st qudit.

In[121]:=
statesQ2 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumPartialTrace\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumPartialTrace\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumPartialTrace"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumPartialTrace"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][#, {1}][
    "NormalizedStateVector"] & /@ m["States"]
Out[121]=

As said, the above quantum circuit should correspond to measurement of Pauli-Y, meaning that the post-measurement states of 2nd qubit should be the same as Pauli-Y eigenstates

Find eigenstates of Pauli-Y:

In[122]:=
basisY = Normal@InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumBasis\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["PauliY"][
    "Elements"] // Reverse
Out[122]=
In[123]:=
basisY == statesQ2
Out[123]=

Quantum Protocols & Algorithms (47) 

Quantum Teleportation (4) 

We shall implement a quantum circuit for teleporting a qubit. The 1st and 2nd qubits represent Alice’s system, while the last one is Bob’s. The double lines carry classical bits:

In[124]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/ace6d6b5-c692-425f-b6c4-ebdf385cc5a3"]
In[125]:=
qc["Diagram"]
Out[125]=

The state to be teleported is =α+β, where α and β are unknown amplitudes. The input state of circuit is as follows:

In[126]:=
\[Psi]0 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumTensorProduct\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumTensorProduct\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumTensorProduct"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumTensorProduct"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{\[Alpha], \[Beta]}], InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], 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121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["PhiPlus"]];

Let’s look at post-measurement states, given the result of Alice’s measurement on 1st and 2nd qubits:

In[127]:=
postMeasurementStates = qc[\[Psi]0]["States"];
Simplify[#["Formula"], Abs[\[Alpha]]^2 + Abs[\[Beta]]^2 == 1] & /@ postMeasurementStates
Out[115]=

As one can see, regardless of measurement results, the state of 3rd qubit is the same state as the original 1st qubit =α+β (with only a normalization difference). Tracing out 1st and 2nd qubit, and compare the reduced state of only qubit 3:

In[128]:=
Thread[InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{\[Alpha], \[Beta]}/
      2] == InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumPartialTrace\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumPartialTrace\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumPartialTrace"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumPartialTrace"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][#, {1, 2}] & /@ postMeasurementStates]
Out[128]=

Deutsch-Jozsa algorithm (2) 

The main goal is to determine if a Boolean function f(x) is constant for all values of x or balanced, where x is a n-digit binary number between 0 and 2n-1, for example when n=2 we have state. We shall consider a balanced implementation of f(x) for n=2 as follows: f(0,0)=0, f(0,1)=f(1,0)=1, f(1,1)=0.

Implement the algorithm:

In[129]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/e5970278-2e50-49ab-ab00-5f353727f101"]
Out[129]=
In[130]:=
qc["Diagram"]
Out[130]=

Action of quantum circuit on the initial state :

In[131]:=
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Out[131]=

As one can see, measuring first two qubits give non-zero 11, indicating a balanced function.

Bernstein-Vazirani Algorithm (1) 

For oracle, given a state we have the following transformation:(-1)s.x, with s being a secret string

In[132]:=
oracle[secret_List] := Module[{list, fs, qubits},
  qubits = Length[secret];
  list = Tuples[{0, 1}, qubits];
  fs = Mod[secret . #, 2] & /@ list;
  InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
   SparseArray[{Table[{i, i}, {i, 2^qubits}] -> (-1)^
       fs}], InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumBasis\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][2, qubits], "Label" -> Rotate["BV oracle", 90 Degree], Range[qubits]]]

an example, lets take the secret string as the 4th state of a 3-qubit system ()

Define the circuit as a list of operators: Hadamards -> Oracle -> Hadamards

In[133]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/12ae7e22-3fce-4ecc-9883-3832bec9f6d2"]

Let’s go through a specific example for 2 qubits (ie n=2) and a secret string 11 (ie the 4th state s=4, meaning )

In[134]:=
BernsteinVazirani[{1, 1}]["Diagram"]
Out[134]=
In[135]:=
final = BernsteinVazirani[{1, 1}][InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"BasisState", {0, 0}}]]["ProbabilityPlot"]
Out[135]=

As one can see, the secret string was and we got it as the final state.

Let's go through another example for 3 qubits and a secret string

In[136]:=
BernsteinVazirani[{1, 0, 0}]["Diagram"]
Out[136]=
In[137]:=
final = BernsteinVazirani[{1, 0, 0}][InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"BasisState", {0, 0, 0}}]]["ProbabilityPlot"]
Out[137]=

As one can see, the section string was the 5th state () and we got it as the final state

Grover's Search Algorithm (5) 

One can describe the Grover’s algorithm as the search for one marked entry which is an n-bit binary number. As the oracle, one can define the following unitary operator:

Define Oracle gate:
In[138]:=
OracleGate[input_List] :=
 Module[{n, matrixRep},
  n = FromDigits[input, 2];
  matrixRep = IdentityMatrix[2^Length[input]];
  matrixRep[[n + 1, n + 1]] = -1;
  InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
   matrixRep, InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumBasis\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumBasis\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumBasis"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][2, Length[input]], "Label" -> Rotate["Grover Oracle", 90 Degree], Range[Length[input]]]]
As an example, define oracle for marking state :
In[139]:=
OracleGate[{1, 0, 1}]
Out[139]=

Check the only element of the operator matrix which is -1 is the same as the number of corresponding binary number.

In[140]:=
n = FromDigits[{1, 0, 1}, 2] + 1;
Position[OracleGate[{1, 0, 1}]["MatrixRepresentation"], -1] == {{n, n}}
Out[141]=

We shall implement 3-qubit Grover algorithm.

Define Grover circuit using a marker state:
In[142]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/6d3d5ec3-093e-4c71-9ee5-e96aeed39878"]
In[143]:=
GroverCircuit[{1, 0, 1}]["Diagram"]
Out[143]=
Define initial state:
In[144]:=
\[Psi]0 = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"BasisState", {0, 0, 0}}];
Return the final state (output of Grover circuit on the initial state)
In[145]:=
\[Psi]f = GroverCircuit[{1, 0, 1}][\[Psi]0];
\[Psi]f["ProbabilityPlot"]
Out[146]=

As one can see, the max probability is for the 5th state, .

Now let’s search for another entry, for example .

In[147]:=
GroverCircuit[{0, 0, 1}][\[Psi]0]["ProbabilityPlot"]
Out[147]=

Search for another entry, for example .

In[148]:=
GroverCircuit[{0, 1, 1}][\[Psi]0]["ProbabilityPlot"]
Out[148]=

Quantum Phase Estimation (7) 

The Quantum Phase Estimation (QPE) algorithm solves the problem of finding θ in the eigenvalue equation U|ψ= e2πiθ|ψ where U is a unitary operator. The input and output of this algorithm are as follows:

INPUT: n+1 qubits at the initial state|ψ〉⊗|0〉⊗n

OUTPUT: |ψ〉⊗|2nθ

Here, we will use QPE to estimate the value of π. This is possible if we choose , which is QantumOperator[{"P",1}], since U|1〉=ei|1〉 and so QPE will return |1〉⊗|2n-1/π. Thus, by measuring the final state, we can estimate the value of π. Below, we will provide a step-by-step example of the algorithm for n = 3.

Construct the QPE circuit:

In[149]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/5467edd0-3b2f-44bb-8136-d8ff0b48d992"]
In[150]:=
qc["Diagram"]
Out[150]=

Initialize the input state

In[151]:=
initState = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"Register", 4}];

Feed the input state to the circuit and extract π from the measurement statistic:

In[152]:=
result = qc[InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"Register", 4}]];
In[153]:=
result["ProbabilityPlot"]
Out[153]=

The distribution peaks at the state 001 which is equivalent to integer 1. So, we have 23-1/π≃1 or π ≃ 4.

In[154]:=
FromDigits[{0, 0, 1}, 2]
Out[154]=
In[155]:=
2^(3 - 1)/%
Out[155]=

Below is the code to build PiEstimationCircuit for any n:

In[156]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/cc129964-b43b-4458-88d5-36ba758e14da"]
In[157]:=
piEstimateQC[5]["Diagram"]
Out[157]=

Let's compute π with 6 qubits (n =5):

In[158]:=
TakeLargest[result1["Probabilities"], 1]
Out[158]=
In[159]:=
FromDigits[{0, 0, 1, 0, 1}, 2]
Out[159]=
In[160]:=
2^(5 - 1)/% // N
Out[160]=

Shor's Algorithm

Shor's algorithm is an algorithm that can factorize a big number using a quantum computer. Given an integer N, Shor's algorithm can return its factors p, q such that N = p × q. This task is difficult for a classical computer if N is large. The algorithm can be summarized in three main steps: classical pre-processing, quantum order-finding, and classical post-processing. Quantum computer is used only on the second step.

1. Classical preprocessing

- If N is even, return 2 and N/2.

- If N is odd, pick a random number 1<a<N

- Compute the greatest-common-divisor (GCD) of a and N. If GCD(a,N ) != 1, return a and N/a.

- If GCD(a, N) = 1, proceed to the order-finding algorithm.

2. Quantum order-finding

- Initialize 2n (for N = 2n-1) qubit register in state |0〉⊗(2n-1)|1〉.

- Apply Hadamard gates H⊗n to the first n qubit.

- Apply modular exponentiation gate U|x〉n|y〉n=|xn|y⊕axmod N〉n where n subscript indicate the number of qubits.

- Apply inverse Quantum Fourier Transform to the first n qubits.

- Measure the first n qubits in computational basis.

3. Classical post-processing

- Let s be the integer representation of the measurement outcome. Compute the order r of axmod N by finding the solutions to j(2n/r) = s where j is an integer.

- If r is odd or ar/2=-1mod N choose a different a and repeat the process.

- Otherwise, at least one of gcd(ar/2+1, N) or gcd(ar/2-1, N) is a non-trivial factor of N.

The quantum circuit used for quantum order-finding can be schematically illustrated as the following (n =3). Note that U4 is the modular exponent gate:

In[161]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/17739016-3e22-4e5c-8f23-049a0843c109"]
Out[161]=

Currently, the specific gates implementation of the modular exponentiation for arbitrary N and a is unknown. However, there are some specific cases that have been studied. Below, we will consider the implementation of quantum order-finding for (N, a) = (15, 7) that has been demonstrated in experiment. Although we need 8 qubits to factor 15 in general, by choosing specific a, it is possible to use only 7 qubits (3 register, 4 ancilla) as demonstrated below.

Quantum order-finding circuit for (N, a) = (15, 7):
In[162]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/5b444ee9-a2e9-4a9e-9694-c55c51c0077f"]
In[163]:=
ShorCircuit1["Diagram"]
Out[163]=

Initialize the qubits state :

In[164]:=
initState = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumTensorProduct\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumTensorProduct\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumTensorProduct"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumTensorProduct"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{"Register", 6}], InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], 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121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumState\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumState\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumState"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumState"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][{0, 1}]}];
Apply the Shor circuit to the initial state:
In[165]:=
result = ShorCircuit1[initState]
Out[165]=
In[166]:=
result["ProbabilityPlot"]
Out[166]=

In decimal notation, the possible states above are: |000〉, |010〉, |100〉, and |110〉. If we obtain |000〉, we need to repeat the protocol. Otherwise,

Outcome: |2〉 j(8/r) = 2 (j, r) = (1, 4)

p=GCD[(74/2+1) ,15] = 5 and q= GCD[(74/2-1),15] = 3

Outcome: |4〉 j(8/r) = 4 (j, r) = (1, 2)

p= GCD[(72/2+1),15] = 1 and q= GCD[(72/2-1),15] = 3

Outcome: |6〉 j(8/r) = 6 (j, r) = (3, 4)

p=GCD[(74/2+1) ,15] = 5 and q= GCD[(74/2-1),15] = 3

Thus, with 0.75 probability, we can find at least one non-trivial factor of 15.

Superdense Coding (3) 

In quantum teleportation we send a single qubit by sending two classical bits over a classical channel. Superdense coding is the reverse process. We send two classical bits using a single qubit and a quantum channel. First, Alice and Bob must sharesa maximally entangled state: where subscripts A and B mean the qubit is held by Alice and Bob respectively. Then, depending on Alice intended message, she performs the following operations to her qubit:

1. If Alice wants to send 00, she does nothing to her qubit.

2. If Alice wants to send 01, she performs X-gate to her qubit.

3. If Alice wants to send 10, she performs Z-gate to her qubit.

4. If Alice wants to send 11, she performs X-gate and then Z-gate to her qubit.

After that, Alice sends her qubit to Bob through a quantum channel. Finally, if now Bob performs a Bell measurement on his qubits, he is guaranteed to receive Alice's message as demonstrated below.

In[167]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/843a979d-0ee1-47c6-865a-aedd7862eebc"]
In[168]:=
superdense["Diagram"]
Out[168]=

Define a initial state as , with xy the code Alice wants to send Bob:

In[169]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/42b10bd5-f529-4122-ac9b-7b8c56641cf5"]

As one can see, for any , Bob will get the same state with 100% probability:

In[170]:=
Multicolumn[
 superdense[state[#]]["ProbabilityPlot"] & /@ Tuples[{0, 1}, 2]]
Out[170]=

Quantum Cryptography (13) 

Cryptography is the study of secure communications. One of the promises of quantum technology is to provide an inherently secure way to send information. The study of exploiting quantum mechanical properties for cryptographic tasks is known as quantum cryptography. One of the well-studied cases of quantum cryptographic protocols is secure quantum key distribution. Before we can understand this protocol, we need to understand a classical communication protocol known as one-time-pad.

In the one-time-pad scenario, Alice wants to send a secret bit string to Bob. To do this securely, they share a common bit key. Alice encodes her message by performing binary addition between her message and the key. She then sends the encoded message through a public channel to Bob. Bob can decode her message by performing the same binary addition. An eavesdropper can not know Alice's message because he does not have access to the key.

Define Alice's message:

In[171]:=
AliceMessage = {0, 1, 0, 0, 0, 0, 1, 1, 0, 1};

Define the shared key:

In[172]:=
SharedKey = {1, 1, 1, 0, 1, 0, 0, 1, 1, 0};

Alice encodes her message by binary addition:

In[173]:=
AliceEncryptedMessage = Mod[#, 2] & /@ (AliceMessage + SharedKey)
Out[173]=

Bob decodes the message by binary addition:

In[174]:=
BobReceivedMessage = Mod[#, 2] & /@ (AliceEncryptedMessage + SharedKey)
Out[174]=

So, using one-time-pad protocol, Alice and Bob can securely communicate. However, we have assumed that no one else has access to the bit key other than Alice and Bob. How can we generate a shared key between Alice and Bob securely? This is the problem of key distribution that can be solved by a quantum BB84 protocol as follows:

1. Alice generates a random bit key and a random basis sequence between Pauli-X and Pauli-Z bases. 2. If the basis is Pauli-Z, she encodes the corresponding bit key in the , basis. If the basis is Pauli-X, she encodes the corresponding bit key in the basis. For example, if her bit keys are 0100 and the bases are XZZX then Alice’s encoding qubits are . 3. Alice sends these qubits to Bob through a quantum channel. 4. Bob also generates a random sequence of basis between Pauli-X and Pauli-Z bases. He then measures Alice's qubits with respect to that basis sequence and keeps the measurement outcome private. 5. Alice and Bob both announce their choice of bases. Then they discard the bits in which their corresponding bases do not match. The remaining bits of Alice and Bob are guaranteed to be perfectly correlated and thus Alice and Bob have shared a key.

Alice generates a bit key and a random basis sequence of length 5:

In[175]:=
AliceBitKey = RandomVariate[CategoricalDistribution[<|0 -> 1/2, 1 -> 1/2|>], 5]
Out[175]=
In[176]:=
AliceRandomBasisChoice = RandomVariate[CategoricalDistribution[<|"Z" -> 1/2, "X" -> 1/2|>], 5]
Out[176]=

Alice encodes her bit key into a 10-qubits state:

In[177]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/5318629a-83ef-48ee-8bd4-4f9ea5fe1ede"]
Out[178]=

Bob Also generate a random basis sequence:

In[179]:=
BobRandomBasisChoice = RandomVariate[CategoricalDistribution[<|"Z" -> 1/2, "X" -> 1/2|>], 5]
Out[179]=

Bob measures Alice's qubits in the chosen basis:

In[180]:=
BobMeasurementCircuit = InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumCircuitOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumCircuitOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumCircuitOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumCircuitOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]][
  Table[Which[
     BobRandomBasisChoice[[i]] == "Z", InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumMeasurementOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumMeasurementOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["PauliZ", {i}][
      "Numeric"], BobRandomBasisChoice[[i]] == "X", InterpretationBox[FrameBox[TagBox[TooltipBox[PaneBox[GridBox[List[List[GraphicsBox[List[Thickness[0.006082377284997087`], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[2.78206`, 85.44900000000004`], List[79.17353999999999`, 41.67850000000004`], List[79.17353999999999`, 1.6779000000000508`], List[2.78206`, 45.44840000000002`]], List[List[127.7826`, 115.44898000000003`], List[127.7826`, 26.449000000000012`], List[161.7832`, 47.44780000000006`], List[161.7832`, 135.44923000000003`]], List[List[5.78267`, 141.44910000000004`], List[40.783`, 121.44819000000003`], List[117.7831`, 165.44919000000004`], List[81.37629999999999`, 187.24592000000004`]]]]], List[FaceForm[RGBColor[0.9843139999999999`, 0.662745`, 0.25098000000000004`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[79.20467000000001`, 90.91309000000003`], List[79.20546`, 48.911800000000056`], List[43.782610000000005`, 69.44830000000003`], List[43.782610000000005`, 111.44764000000002`]], List[List[82.77993`, 137.44809000000004`], List[48.77938`, 117.44782000000004`], List[82.77993`, 97.44923000000004`], List[117.7808`, 117.44782000000004`]], List[List[86.41823000000001`, 90.97696000000002`], List[86.45015000000001`, 48.71930000000003`], List[121.7829`, 70.44880000000003`], List[121.7829`, 112.44819000000003`]]]]], List[FaceForm[RGBColor[0.46274499999999996`, 0.066667`, 0.07451`, 1.`]]], Rule[StripOnInput, False]], StyleBox[List[FilledCurveBox[List[List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]], List[List[0, 2, 0], List[0, 1, 0], List[0, 1, 0]]], List[List[List[158.7825`, 141.44910000000004`], List[123.7817`, 121.44886000000004`], List[89.78278999999999`, 141.44910000000004`], List[123.7817`, 161.44770000000003`]], List[List[2.7817600000000002`, 92.44794000000003`], List[37.782610000000005`, 72.44770000000003`], List[37.782610000000005`, 114.44870000000003`], List[2.7817600000000002`, 134.44898000000003`]], List[List[86.45015000000001`, 1.7736000000000445`], List[121.78269999999999`, 23.44920000000002`], List[121.78269999999999`, 63.33730000000003`], List[86.45015000000001`, 41.66350000000003`]]]]], List[FaceForm[RGBColor[1.`, 0.41568600000000006`, 0.054902000000000006`, 1.`]]], Rule[StripOnInput, False]]], List[Rule[BaselinePosition, Scaled[0.15`]], Rule[ImageSize, 10], Rule[ImageSize, List[30, Automatic]]]], StyleBox[RowBox[List["\"QuantumMeasurementOperator\"", " "]], Rule[ShowAutoStyles, False], Rule[ShowStringCharacters, False], Rule[FontSize, Times[0.9`, Inherited]], Rule[FontColor, GrayLevel[0.1`]]]]], Rule[GridBoxSpacings, List[Rule["Columns", List[List[0.25`]]]]]], Rule[Alignment, List[Left, Baseline]], Rule[BaselinePosition, Baseline], Rule[FrameMargins, List[List[3, 0], List[0, 0]]], Rule[BaseStyle, List[Rule[LineSpacing, List[0, 0]], Rule[LineBreakWithin, False]]]], RowBox[List["PacletSymbol", "[", RowBox[List["\"Wolfram: QuantumFramework\"", ",", "\"QuantumMeasurementOperator\""]], "]"]], Rule[TooltipStyle, List[Rule[ShowAutoStyles, True], Rule[ShowStringCharacters, True]]]], Function[Annotation[Slot[1], Style[Defer[PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"]], Rule[ShowStringCharacters, True]], "Tooltip"]]], Rule[Background, RGBColor[0.968`, 0.976`, 0.984`]], Rule[BaselinePosition, Baseline], Rule[DefaultBaseStyle, List[]], Rule[FrameMargins, List[List[0, 0], List[1, 1]]], Rule[FrameStyle, RGBColor[0.831`, 0.847`, 0.85`]], Rule[RoundingRadius, 4]], PacletSymbol["Wolfram: QuantumFramework", "QuantumMeasurementOperator"], Rule[Selectable, False], Rule[SelectWithContents, True], Rule[BoxID, "PacletSymbolBox"]]["PauliX", {i}]][
    "Numeric"], {i, 5}]]
Out[180]=

Let's suppose Bob obtains the following outcome:

In[181]:=
Select[BobMeasurementCircuit[keyQubits]["Probabilities"], # != 0 &]
Out[181]=
In[182]:=
BobBitKey = RandomVariate[
    BobMeasurementCircuit[keyQubits]["Distribution"]]["Name"] /. {
\!\(\*SubscriptBox[\("\[Psi]"\), \("z-"\)]\) | 
\!\(\*SubscriptBox[\("\[Psi]"\), \("x-"\)]\) -> 0, 
\!\(\*SubscriptBox[\("\[Psi]"\), \("z+"\)]\) | 
\!\(\*SubscriptBox[\("\[Psi]"\), \("x+"\)]\) -> 1}
Out[182]=

After Bob obtains his result, Alice and Bob announce their basis choice publicly and decide which bits are to be discarded:

In[183]:=
discardIndices = Table[If[! BobRandomBasisChoice[[i]] == AliceRandomBasisChoice[[i]], i, ## &[]], {i, Length[AliceRandomBasisChoice]}]
Out[183]=

Alice's final key:

In[184]:=
AliceFinalKey = Delete[AliceBitKey, {#} & /@ discardIndices]
Out[184]=

Bob's final key:

In[185]:=
BobFinalKey = Delete[BobBitKey, {#} & /@ discardIndices]
Out[185]=

In this particular implementation, Alice and Bob successfully share a 4-bits key. They can repeat the procedure until the key has the same length as the length of the intended message. This protocol is secure because prior to the bases choice's public announcement, an eavesdropper can only extract the keys by guessing the measurement basis. However, by doing this he risks disturbing the state. So, to detect an eavesdropper, Alice and Bob can sacrifice some of their key bits to make sure that their bits perfectly correlate. If they do not correlate, this means Alice and Bob have detected the presence of an eavesdropper.

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