# Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

This resource function is obsolete. Use the paclet Wolfram/QuantumFramework instead.

Function Repository Resource:

Represent a discrete quantum circuit operator

Contributed by:
Jonathan Gorard (with modifications by Ruhi Shah and Taufiq Murtadho)

ResourceFunction["QuantumCircuitOperator"][{ represents a discrete quantum circuit operator with gates (i.e. QuantumDiscreteOperator or QuantumMeasurementOperator objects) | |

ResourceFunction["QuantumCircuitOperator"][ transforms a specified ResourceFunction["QuantumCircuitOperator"] | |

ResourceFunction["QuantumCircuitOperator"][ transforms |

The QuantumBasis object is produced by the resource function QuantumBasis. When a QuantumBasis object is not specified, the default is taken to be QuantumBasis["Computational"].

When an order *order* is not specified, the default is taken to be Range[*arity*], where *arity* designates the arity of the ResourceFunction["QuantumCircuitOperator"] (i.e. as computed from the arities of the constituent gates).

The permitted values of *pic* in ResourceFunction["QuantumCircuitOperator"][…,*pic*] are "Schrodinger" (default - may also be written as "Schroedinger" or "Schrödinger"), "Heisenberg", "Interaction" and "PhaseSpace".

In ResourceFunction["QuantumCircuitOperator"][{*gate*_{1},*gate*_{2},…}], the *gate*_{i} represent either QuantumDiscreteOperator or QuantumMeasurementOperator objects. These gates are applied sequentially (or, whenever the wires do not overlap, in parallel) based on the order of the list. By convention, QuantumDiscreteOperator gates should be placed towards the beginning of the circuit and QuantumMeasurementOperator towards the end.

ResourceFunction["QuantumCircuitOperator"][…]["*prop*"] gives the property "*prop*" of the specified discrete quantum circuit operator. Possible properties include:

"Operator" | association <|b_{11}→e_{11},b_{12}→e_{12},…|> of tensor products of basis names and matrix elements |

"Basis" | which QuantumBasis the circuit operator is defined with respect to |

"Picture" | which quantum mechanical picture the circuit operator is defined with respect to |

"BasisElements" | list of basis elements b_{i} |

"MatrixRepresentation" | matrix representation mat of the circuit operator |

"OrderedMatrixRepresentation" | matrix representation adjusted according to the order of the circuit operator, assuming an input state containing Max[order] qudits (subsystems) |

{"OrderedMatrixRepresentation",n} | matrix representation adjusted according to the order of the circuit operator, assuming an input state containing n qudits (subsystems) |

"OperatorRepresentation" | representation of the circuit operator as a QuantumDiscreteOperator |

"Arity" | arity of the circuit operator |

"Dimensions" | dimensionality of each qudit (subsystem) in the input state for the circuit operator |

"Order" | order in which the circuit operator should be applied to qudits (subsystems) in the input state |

"HermitianQ" | whether the circuit operator is Hermitian (self-adjoint) |

"UnitaryQ" | whether the circuit operator is unitary (an automorphism of Hilbert space) |

"Eigenvalues" | eigenvalues of the matrix representation of the circuit operator |

"Eigenvectors" | eigenvectors of the matrix representation of the circuit operator |

"Gates" | list of gates (QuantumDiscreteOperator and QuantumMeasurementOperator objects) contained within the circuit operator |

"GateOperators" | list of associations <|b_{11}→e_{11},b_{12}→e_{12},…|> of tensor products of basis names and matrix elements for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GateBases" | list of QuantumBasis objects for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GatePictures" | list of quantum mechanical pictures for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GateBasisElements" | list of lists of basis elements b_{i} for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GateMatrixRepresentations" | list of matrix representations mat for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GateOrderedMatrixRepresentations" | list of ordered matrix representations for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit, assuming an input state containing Max[Max/@orders] qudits (subsystems) |

{"GateOrderedMatrixRepresentations",n} | list of ordered matrix representations for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit, assuming an input state containing n qudits (subsystems) |

"GateArities" | list of arities for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GateDimensions" | list of qudit dimensionalities for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"GateOrders" | list of qudit application orders for each QuantumDiscreteOperator or QuantumMeasurementOperator in the circuit |

"Diagram" | diagrammatic representation of the circuit operator |

"Properties" | list of all property names |

A ResourceFunction["QuantumCircuitOperator"] (in the Schrodinger picture) can act on a QuantumDiscreteState (also in the Schrodinger picture) using ResourceFunction["QuantumCircuitOperator"][…][QuantumDiscreteState[…]]. Likewise for ResourceFunction["QuantumCircuitOperator"] objects acting on QuantumDiscreteOperator, QuantumMeasurementOperator or QuantumHamiltonianOperator objects in the Heisenberg picture, etc.

When a ResourceFunction["QuantumCircuitOperator"] acts on a QuantumDiscreteState, it will either return another QuantumDiscreteState object (if the circuit contains no measurement gates) or a QuantumMeasurement object (if the circuit contains measurement gates).

A QuantumDiscreteState, QuantumDiscreteOperator, QuantumMeasurementOperator, QuantumMeasurement or QuantumHamiltonianOperator object can be created with the corresponding resource functions.

The following named QuantumDiscreteOperator objects come equipped with their own special gate symbols in the circuit diagram representation (in some, though not all, cases, the name of the operator is also the symbol shown within the gate):

"S" | S-gate (equivalent to a π/2 PHASE gate) |

"T" | T-gate (equivalent to a π/4 PHASE gate) |

"CNOT" | CNOT gate for qubits (controlled NOT - equivalent to a CX gate) |

{"CNOT",d} | CNOT gate for d-dimensional qudits |

"CPHASE" | CPHASE gate for qubits (controlled π PHASE gate) |

{"CPHASE",d} | CPHASE gate for d-dimensional qudits |

"CX" | CX gate for qubits (controlled Pauli-X gate - equivalent to a CNOT gate) |

{"CX",d} | CX gate for d-dimensional qudits |

"CY" | CY gate for qubits (controlled Pauli-Y gate) |

"CZ" | CZ gate for qubits (controlled Pauli-Z gate) |

{"CZ",d} | CZ gate for d-dimensional qudits |

"Fourier" | QFT gate for qubits (quantum Fourier transform) |

{"Fourier",d} | QFT gate for d-dimensional qudits |

"InverseFourier" | IQFT gate for qubits (inverse quantum Fourier transform) |

{"InverseFourier",d} | IQFT gate for d-dimensional qudits |

"SWAP" | SWAP gate for qubits (swap two qubits) |

{"SWAP",d} | SWAP gate for d-dimensional qudits |

"RootSWAP" | square root of the SWAP gate for qubits (half swap two qubits) |

{"RootSWAP",d} | square root of the SWAP gate for d-dimensional qudits |

"SUM" | SUM gate for qubits (half-adder for qubits - equivalent to a CNOT gate) |

{"SUM",d} | SUM gate for d-dimensional qudits |

"PauliX" | Pauli-X gate for qubits (equivalent to a π rotation around the x axis of the Bloch sphere) |

{"PauliX",d} | Pauli-X gate for d-dimensional qudits |

"PauliY" | Pauli-Y gate for qubits (equivalent to a π rotation around the y axis of the Bloch sphere) |

"PauliZ" | Pauli-Z gate for qubits (equivalent to a π rotation around the z axis of the Bloch sphere) |

{"PauliZ",d} | Pauli-Z gate for d-dimensional qudits |

"RootNOT" | square root of the Pauli-X gate for qubits (half NOT operation on one qubit) |

{"RootNOT",d} | square root of the Pauli-X gate for d-dimensional qudits |

"Hadamard" | Hadamard gate for qubits (equivalent to a QFT gate on one qubit) |

"Toffoli" | Toffoli gate on 3 qubits (equivalent to a π/2 Deutsch gate) |

{"Toffoli",n} | Toffoli gate on n qubits |

"CSWAP" | controlled SWAP gate for qubits (Fredkin gate) |

The following named QuantumMeasurementOperator objects come equipped with their own special gate symbols in the circuit diagram representation (in some, though not all, cases, the name of the operator is also the symbol shown within the gate):

"PauliX" | measurement operator based on the Pauli-X gate for qubits (equivalent to a π rotation around the x axis of the Bloch sphere) |

{"PauliX",d} | measurement operator based on the Pauli-X gate for d-dimensional qudits (not Hermitian for general d>2) |

"PauliY" | measurement operator based on the Pauli-Y gate for qubits (equivalent to a π rotation around the y axis of the Bloch sphere) |

"PauliZ" | measurement operator based on the Pauli-Z gate for qubits (equivalent to a π rotation around the z axis of the Bloch sphere) |

{"PauliZ",d} | measurement operator based on the Pauli-Z gate for d-dimensional qudits (not Hermitian for general d>2) |

"Hadamard" | measurement operator based on the Hadamard gate for qubits (equivalent to a QFT gate on one qubit) |

"ComputationalBasis" | measurement in the 2-dimensional computational basis |

{"ComputationalBasis",d} | measurement in the d-dimensional computational basis |

{"ComputationalBasis",d,eigs} | measurement in the d-dimensional computational basis with eigenvalues (i.e. measurement outcomes) eigs |

"BellBasis" | measurement in the Bell basis |

{"BellBasis",eigs} | measurement in the Bell basis with eigenvalues (i.e. measurement outcomes) eigs |

"PauliBasis" | measurement in the basis of Pauli matrices |

{"PauliBasis",eigs} | measurement in the basis of Pauli matrices with eigenvalues (i.e. measurement outcomes) eigs |

"FourierBasis" | measurement in the 2-dimensional basis of the quantum Fourier transform |

{"FourierBasis",d} | measurement in the d-dimensional basis of the quantum Fourier transform |

{"FourierBasis",d,eigs} | measurement in the d-dimensional basis of the quantum Fourier transform with eigenvalues (i.e. measurement outcomes) eigs |

All other QuantumDiscreteOperator and QuantumMeasurementOperator objects not listed above are rendered using generic gate symbols.

Create a discrete quantum circuit operator in the computational basis consisting of a Hadamard gate and a Pauli-Z gate applied in series:

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Show the diagrammatic representation:

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Apply the circuit operator to the first qubit (default) of a two-qubit discrete quantum state in the computational basis:

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Apply the circuit operator to the second qubit of the state instead:

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Apply the Hadamard and Pauli-Z gates in parallel to qubits 1 and 2 instead:

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Create a discrete quantum circuit operator in the computational basis consisting of a CNOT gate with control qubit 2 and "active" qubit 3, followed by a SWAP gate for qubits 1 and 3, followed by a quantum Fourier transform gate for qubits 1, 2 and 3:

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Return its matrix representation:

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Return the ordered matrix representation (assuming a state of size 4):

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Return the new ordered matrix representation (assuming a state of size 4):

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Create a discrete quantum circuit operator in the computational basis consisting of an inverse quantum Fourier transform gate for qubits 1 and 2, followed by a computational basis measurement gate for qubit 2:

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By default, all discrete quantum circuit operators are created in the Schrodinger picture:

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Circuit operators in the Schrodinger picture can be applied to QuanumDiscreteState objects:

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However, they cannot be applied to other operators (i.e. QuantumDiscreteOperator objects, QuantumMeasurementOperator objects or QuantumHamiltonianOperator objects). Doing so will return unevaluated:

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On the other hand, circuit operators created in the Heisenberg picture cannot be applied to QuantumDiscreteState objects (will return unevaluated):

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However, they can be applied to other operators in the Heisenberg picture (i.e. QuantumDiscreteOperator objects, QuantumMeasurementOperator objects or QuantumHamiltonianOperator objects):

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Apply a discrete quantum circuit operator (without measurement gates) to a QuantumMeasurement object:

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Show the resulting (evolved) states:

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Apply a discrete quantum circuit operator (with measurement gates) to a QuantumMeasurement object:

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Show the resulting joint distribution:

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Create a discrete quantum circuit operator in the computational basis (default):

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Transform the circuit operator to the Fourier basis:

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Transform the circuit operator back to the computational basis:

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Represent a discrete quantum circuit operator in the Heisenberg picture:

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Transform the circuit to the interaction picture:

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Represent a two-qubit discrete quantum circuit operator in the computational basis:

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Represent the same circuit operator in the Pauli-X basis instead:

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Circuit operators in the Schrodinger picture can be applied to QuantumDiscreteState objects that are also in the Schrodinger picture:

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However, they cannot be applied to other operators in the Schrodinger picture (returns unevaluated):

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Likewise, circuit operators in the "PhaseSpace" picture can be applied to QuantumDiscreteState objects in the "PhaseSpace" picture:

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But they cannot be applied to other operators in the phase space picture (returns unevaluated):

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On the other hand, circuit operators in the Heisenberg picture cannot be applied to QuantumDiscreteState objects in the Heisenberg picture:

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But they can be applied to other operators in the Heisenberg picture:

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Finally, circuit operators in the Interaction picture can be applied to both QuantumDiscreteState objects in the Interaction picture:

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And also to other operators in the Interaction picture:

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In general, the quantum mechanical picture of a QuantumCircuitOperator and the object that it is being applied to must match.

QuantumCircuitOperator objects can be composed by application:

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Circuits can also be composed by including them as "gates" in the specification of other circuits:

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Show the circuit diagram for an S-gate (a *π*/2 phase change gate):

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Show the circuit diagram for an S-gate with a control qubit:

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Show the circuit diagram for a T-gate (a *π*/4 phase change gate):

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Show the circuit diagram for a T-gate with a control qubit:

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Show the circuit diagram for a CNOT gate (a controlled NOT gate—equivalent to a controlled Pauli-X gate):

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Show the circuit diagram for a Pauli-X gate (a *π* rotation around the *x* axis of the Bloch sphere):

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Show the circuit diagram for a Pauli-Y gate (a *π* rotation around the *y* axis of the Bloch sphere):

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Show the circuit diagram for a controlled Pauli-Y gate:

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Show the circuit diagram for a Pauli-Z gate (a *π* rotation around the *z* axis of the Bloch sphere):

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Show the circuit diagram for a controlled Pauli-Z gate:

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Show the circuit diagram for a two-qubit QFT gate (quantum Fourier transform):

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Show the circuit diagram for a two-qubit QFT gate with a control qubit:

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Show the circuit diagram for a two-qubit IQFT gate (inverse quantum Fourier transform):

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Show the circuit diagram for a two-qubit IQFT gate with a control qubit:

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Show the circuit diagram for a SWAP gate:

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Show the circuit diagram for a SWAP gate with a control qubit:

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Show the circuit diagram for the square root of a SWAP gate (half swaps two qubits):

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Show the circuit diagram for the square root of a SWAP gate with a control qubit:

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Show the circuit diagram for a Hadamard gate (a single-qubit quantum Fourier transform):

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Show the circuit diagram for a Hadamard gate with a control qubit:

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Show the circuit diagram for a positive operator-valued discrete quantum measurement:

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Show the circuit diagram for a Pauli-X measurement gate (a *π* rotation around the *x* axis of the Bloch sphere):

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Show the circuit diagram for a Pauli-Y measurement gate (a *π* rotation around the *y* axis of the Bloch sphere):

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Show the circuit diagram for a Pauli-Z measurement gate (a *π* rotation around the *z* axis of the Bloch sphere):

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Show the circuit diagram for a Hadamard measurement gate (a single-qubit quantum Fourier transform):

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Show the circuit diagram for a single-qubit computational basis measurement (with default eigenvalues):

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Show the circuit diagram for a two-qubit computational basis measurement (with default eigenvalues):

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Show the circuit diagram for a single-qubit Bell basis measurement (with default eigenvalues):

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Show the circuit diagram for a two-qubit Bell basis measurement (with default eigenvalues):

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Show the circuit diagram for a single-qubit Fourier basis measurement:

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Show the circuit diagram for a two-qubit Fourier basis measurement:

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View a list of properties that can be extracted from a QuantumCircuitOperator object:

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Return the circuit operator association:

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Return which QuantumBasis the circuit operator is defined with respect to:

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Return which quantum mechanical picture the circuit operator is defined with respect to:

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Return the association of names and basis elements:

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Return the matrix representation:

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Return the ordered matrix representation (when applied to qubits 2 and 3), assuming a state consisting of 3 qubits:

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Return the ordered matrix representation (when applied to qubits 2 and 3), assuming a state consisting of 4 qubits:

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Return the representation as a QuantumDiscreteOperator:

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Return the arity (number of input subsystems):

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Return the number of dimensions:

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Return the order in which the circuit operator is applied to subsystems:

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Determine whether the circuit operator is Hermitian:

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Determine whether the circuit operator is unitary:

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Return the list of eigenvalues:

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Return the list of eigenvectors:

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Return the list of gates in the circuit:

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Return the list of operator associations for each gate:

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Return the list of QuantumBasis objects for each gate:

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Return the list of quantum mechanical pictures for each gate:

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Return the list of associations of names and basis elements for each gate:

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Return the list of matrix representations for each gate:

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Return the list of ordered matrix representations for each gate, assuming a state consisting of 3 qubits:

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Return the list of ordered matrix representations for each gate, assuming a state consisting of 4 qubits:

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Return the list of arities (number of input subsystems) for each gate:

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Return the list of numbers of dimensions for each gate:

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Return the list of orders in which the gates are applied to subsystems:

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Show the diagrammatic representation of the circuit operator:

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Represent a three-qubit quantum Fourier transform as a composition of Hadamard gates, controlled S-phase and T-phase gates and a single SWAP operation at the end:

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Apply the circuit operator to a random three-qubit discrete quantum state:

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Compare the matrix representation of the circuit operator to the matrix representation of the standard three-qubit quantum Fourier transform:

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Compare the effect of the standard three-qubit quantum Fourier transform on the random state from a previous example:

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Represent a Hadamard test circuit (for creating a random variable whose expected value is the expected real part of the observed value of a quantum state with respect to some unitary operator):

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Apply the circuit operator to a random three-qubit discrete quantum state:

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Represent a quantum parity check circuit (for determining the expected number of 1s in an arbitrary string of qubits), shown here for the three-qubit case:

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Apply the circuit operator to a random four-qubit discrete quantum state:

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- 1.0.0 – 01 June 2021

This work is licensed under a Creative Commons Attribution 4.0 International License