Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
Quantum object abstraction
a nice formatting for qudit names | |
an association for qudit names and their corresponding tensor representation | |
[<|"Input" QuditBasis[…],"Output" QuditBasis[…],"Picture"…,"Label"…,"Parameters"…|>] | input and output, and their corresponding qudit basis |
state representation in a given quantum basis | |
quantum state with order | |
quantum operator with target | |
a wrapper for quantum measurement operator | |
a wrapper for quantum operator | |
a wrapper for a list of quantum operators and their composition |
We will briefly discuss how the abstraction is done in our framework and how quantum objects depend on each other.
QuditName
A qudit name, as Φ, in bra format:
In[57]:=
 | QuditName |
Out[57]=
〈Φ|
A qudit name, as Φ, in ket format:
| QuditName |
|Φ〉
1D qudit does not represent a physical qudit, and we will use it as dummy qudit. We should also consider the cases of no qudit, which is defined as follows
| QuditName |
A zero-dimension qudit is also possible:
| QuditName |
∅
We will explain in the following sections why it is important to have a well-defined no-qudit case.
QuditBasis
The next level of abstraction is , which is a wrapper for an association with keys as names and values as the corresponding tensor representation.
The key-value has this form:, where i enumerates qudits (to describe many qudit systems).
The key-value has this form:
{QuditName[…],i}tensor
MatrixForm/@
[2]["Representations"]
 | QuditBasis |
{|0〉,1}
,{|1〉,1}
1 |
0 |
0 |
1 |
MatrixForm/@
[3]["Representations"]
| QuditBasis |
{|0〉,1}
,{|1〉,1}
,{|2〉,1}
1 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
MatrixForm/@
[{"I","X"}]["Representations"]
| QuditBasis |
{|0〉,1}
,{|1〉,1}
,{|〉,2}
,{|〉,2}
1 |
0 |
0 |
1 |
+
1 2 |
1 2 |
−
1 2 |
- 1 2 |
The actual basis for the above example is generated by the tensor product of above tensors:
MatrixForm/@
[{"I","X"}]["Association"]
| QuditBasis |
0
,0
,1
,1
+
1 2 | 1 2 |
0 | 0 |
−
1 2 | - 1 2 |
0 | 0 |
+
0 | 0 |
1 2 | 1 2 |
−
0 | 0 |
1 2 | - 1 2 |
For example, the first element of above association is generated as
TensorProduct{1,0},{1,1}
2
//MatrixForm1 2 | 1 2 |
0 | 0 |
Unit-basis is defined as 1D-qudit case:
| QuditBasis |
QuditBasis
 |  |
|
QuantumBasis
QuantumBasis
The next level of abstraction is our framework is . Its generic form is as follows: .
In contrast to previous quantum objects, the key additions for are the input and output information. All other details are categorized as the "Meta" data.
 | QuantumBasis |
{PictureSchrödinger,LabelNone,ParameterSpec{}}
Pauli-X quantum basis:
| QuantumBasis |
QuantumBasis[<|Input -> QuditBasis[
<|{QuditName[I., Dual -> False], 1} ->
1|>], Output -> QuditBasis[
<|{QuditName[0, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{1}}}, {1}}],
{QuditName[1, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{2}}}, {1}}]|>],
Picture -> Schrödinger, Label -> I,
ParameterSpec -> {}|>]
<|{QuditName[I., Dual -> False], 1} ->
1|>], Output -> QuditBasis[
<|{QuditName[0, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{1}}}, {1}}],
{QuditName[1, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{2}}}, {1}}]|>],
Picture -> Schrödinger, Label -> I,
ParameterSpec -> {}|>]
Note in this case, there is no input, and only one output:
| QuantumBasis |
QuditBasis
|  |
|
| QuantumBasis |
QuditBasis
|  |
|
The diagram for the basis shows clearly the input and output:
| QuantumBasis |
In the Wolfram Quantum Framework, the bold formatting in the summary box signifies input, corresponding to column indices (covariant, or bra), whereas the normal formatting represents output, corresponding to row indices (contravariant, or ket). An illustration of this distinction can be found in the summary box of the following basis:
| QuantumBasis |
QuantumBasis
 |  |
|
QuantumState
The next level of abstraction is , which has the general form of , with state can be a state vector or a density matrix.
state=
[{1,,1,,3,3},{2,3}];state//InputForm
| QuantumState |
QuantumState[SparseArray[Automatic, {6}, 0,
{1, {{0, 6}, {{1}, {2}, {3}, {4}, {5}, {6}}},
{1, I, 1, I, 3, 3*I}}],
QuantumBasis[<|Input -> QuditBasis[
<|{QuditName[I., Dual -> False], 1} ->
1|>], Output -> QuditBasis[
<|{QuditName[0, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{1}}}, {1}}],
{QuditName[1, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{2}}}, {1}}],
{QuditName[0, Dual -> False], 2} ->
SparseArray[Automatic, {3}, 0,
{1, {{0, 1}, {{1}}}, {1}}],
{QuditName[1, Dual -> False], 2} ->
SparseArray[Automatic, {3}, 0,
{1, {{0, 1}, {{2}}}, {1}}],
{QuditName[2, Dual -> False], 2} ->
SparseArray[Automatic, {3}, 0,
{1, {{0, 1}, {{3}}}, {1}}]|>],
Picture -> Schrödinger,
Label -> None, ParameterSpec -> {}|>]]
{1, {{0, 6}, {{1}, {2}, {3}, {4}, {5}, {6}}},
{1, I, 1, I, 3, 3*I}}],
QuantumBasis[<|Input -> QuditBasis[
<|{QuditName[I., Dual -> False], 1} ->
1|>], Output -> QuditBasis[
<|{QuditName[0, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{1}}}, {1}}],
{QuditName[1, Dual -> False], 1} ->
SparseArray[Automatic, {2}, 0,
{1, {{0, 1}, {{2}}}, {1}}],
{QuditName[0, Dual -> False], 2} ->
SparseArray[Automatic, {3}, 0,
{1, {{0, 1}, {{1}}}, {1}}],
{QuditName[1, Dual -> False], 2} ->
SparseArray[Automatic, {3}, 0,
{1, {{0, 1}, {{2}}}, {1}}],
{QuditName[2, Dual -> False], 2} ->
SparseArray[Automatic, {3}, 0,
{1, {{0, 1}, {{3}}}, {1}}]|>],
Picture -> Schrödinger,
Label -> None, ParameterSpec -> {}|>]]
The above state is a conventional quantum state (a composite 2⊗3 dimensional state), with two outputs and no input:
state["Diagram"]
Show input:
Show output:
Create a bra state, which is the conjugate-transpose of the original state:
A state with no output and only input is called an effect (see the above summary box).
As one can see, a bra has no output, but only input:
Return the input:
Return the output:
An inner product (bra-ket):
An inner product is a scalar (see the summary box above):
The inner product is a pure scalar (no input, no output):
Create a ket-bra:
Return the traditional form:
A ket-bra is called map, which has both input and output:
Show diagram:
Show corresponding quantum basis
QuantumOperator
The next level of abstraction in our framework is QuantumOperator, which is QuantumOperator[QuantumState[...], order]. In other words, one can see it as a quantum state plus an order, where the order describes where the operator acts.
Show input form of Pauli-x operator, acting on 2nd qubit
Show diagram of a Pauli-X operator acting on 2nd qubit
Show input order
Show output order
QuantumMeasurementOperator
QuantumMeasurement
It is a wrapper for quantum measurement operator, that represents a quantum measurement
QuantumChannel
It is a wrapper for quantum operators, representing a quantum channel
QuantumCircuitOperator
QuantumCircuitOperator is abstracted as an association with elements as list of operators, ops, and a circuit label, i.e. QuantumCircuitOperator[<|”Elements” -> ops, “Label” -> “Bell”|>]