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# qiskit.circuit.library.RZXGate¶

class RZXGate(theta)[source]

A parameteric 2-qubit $$Z \otimes X$$ interaction (rotation about ZX).

This gate is maximally entangling at $$\theta = \pi/2$$.

The cross-resonance gate (CR) for superconducting qubits implements a ZX interaction (however other terms are also present in an experiment).

Circuit Symbol:

     ┌─────────┐
q_0: ┤0        ├
│  Rzx(θ) │
q_1: ┤1        ├
└─────────┘


Matrix Representation:

\begin{align}\begin{aligned}\newcommand{\th}{\frac{\theta}{2}}\\\begin{split}R_{ZX}(\theta)\ q_0, q_1 = exp(-i \frac{\theta}{2} X{\otimes}Z) = \begin{pmatrix} \cos(\th) & 0 & -i\sin(\th) & 0 \\ 0 & \cos(\th) & 0 & i\sin(\th) \\ -i\sin(\th) & 0 & \cos(\th) & 0 \\ 0 & i\sin(\th) & 0 & \cos(\th) \end{pmatrix}\end{split}\end{aligned}\end{align}

Note

In Qiskit’s convention, higher qubit indices are more significant (little endian convention). In the above example we apply the gate on (q_0, q_1) which results in the $$X \otimes Z$$ tensor order. Instead, if we apply it on (q_1, q_0), the matrix will be $$Z \otimes X$$:

     ┌─────────┐
q_0: ┤1        ├
│  Rzx(θ) │
q_1: ┤0        ├
└─────────┘

\begin{align}\begin{aligned}\newcommand{\th}{\frac{\theta}{2}}\\\begin{split}R_{ZX}(\theta)\ q_1, q_0 = exp(-i \frac{\theta}{2} Z{\otimes}X) = \begin{pmatrix} \cos(\th) & -i\sin(\th) & 0 & 0 \\ -i\sin(\th) & \cos(\th) & 0 & 0 \\ 0 & 0 & \cos(\th) & i\sin(\th) \\ 0 & 0 & i\sin(\th) & \cos(\th) \end{pmatrix}\end{split}\end{aligned}\end{align}

This is a direct sum of RX rotations, so this gate is equivalent to a uniformly controlled (multiplexed) RX gate:

$\begin{split}R_{ZX}(\theta)\ q_1, q_0 = \begin{pmatrix} RX(\theta) & 0 \\ 0 & RX(-\theta) \end{pmatrix}\end{split}$

Examples:

$R_{ZX}(\theta = 0) = I$
$R_{ZX}(\theta = 2\pi) = -I$
$R_{ZX}(\theta = \pi) = -i Z \otimes X$
$\begin{split}RZX(\theta = \frac{\pi}{2}) = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 0 & -i & 0 \\ 0 & 1 & 0 & i \\ -i & 0 & 1 & 0 \\ 0 & i & 0 & 1 \end{pmatrix}\end{split}$

Create new RZX gate.

__init__(theta)[source]

Create new RZX gate.

Methods

 __init__(theta) Create new RZX gate. add_decomposition(decomposition) Add a decomposition of the instruction to the SessionEquivalenceLibrary. Assemble a QasmQobjInstruction broadcast_arguments(qargs, cargs) Validation and handling of the arguments and its relationship. c_if(classical, val) Add classical condition on register classical and value val. control([num_ctrl_qubits, label, ctrl_state]) Return controlled version of gate. copy([name]) Copy of the instruction. Return inverse RZX gate (i.e. Return True .IFF. DEPRECATED: use instruction.reverse_ops(). power(exponent) Creates a unitary gate as gate^exponent. Return a default OpenQASM string for the instruction. Creates an instruction with gate repeated n amount of times. For a composite instruction, reverse the order of sub-instructions. Return a numpy.array for the RZX gate. validate_parameter(parameter) Gate parameters should be int, float, or ParameterExpression

Attributes

 decompositions Get the decompositions of the instruction from the SessionEquivalenceLibrary. definition Return definition in terms of other basic gates. duration Get the duration. label Return gate label params return instruction params. unit Get the time unit of duration.
add_decomposition(decomposition)

Add a decomposition of the instruction to the SessionEquivalenceLibrary.

assemble()

Assemble a QasmQobjInstruction

Return type

Instruction

broadcast_arguments(qargs, cargs)

Validation and handling of the arguments and its relationship.

For example, cx([q,q], q) means cx(q, q); cx(q, q). This method yields the arguments in the right grouping. In the given example:

in: [[q,q], q],[]
outs: [q, q], []
[q, q], []


• If len(qargs) == 1:

[q, q] -> [q],[q]

• If len(qargs) == 2:

[[q, q], [r, r]] -> [q, r], [q, r]
[[q], [r, r]]       -> [q, r], [q, r]
[[q, q], [r]]       -> [q, r], [q, r]

• If len(qargs) >= 3:

[q, q], [r, r],  ...] -> [q, r, ...], [q, r, ...]

Parameters
• qargs (List) – List of quantum bit arguments.

• cargs (List) – List of classical bit arguments.

Return type

Tuple[List, List]

Returns

A tuple with single arguments.

Raises

CircuitError – If the input is not valid. For example, the number of arguments does not match the gate expectation.

c_if(classical, val)

Add classical condition on register classical and value val.

control(num_ctrl_qubits=1, label=None, ctrl_state=None)

Return controlled version of gate. See ControlledGate for usage.

Parameters
• num_ctrl_qubits (Optional[int]) – number of controls to add to gate (default=1)

• label (Optional[str]) – optional gate label

• ctrl_state (Union[int, str, None]) – The control state in decimal or as a bitstring (e.g. ‘111’). If None, use 2**num_ctrl_qubits-1.

Returns

Controlled version of gate. This default algorithm uses num_ctrl_qubits-1 ancillae qubits so returns a gate of size num_qubits + 2*num_ctrl_qubits - 1.

Return type

qiskit.circuit.ControlledGate

Raises

QiskitError – unrecognized mode or invalid ctrl_state

copy(name=None)

Copy of the instruction.

Parameters

name (str) – name to be given to the copied circuit, if None then the name stays the same.

Returns

a copy of the current instruction, with the name

updated if it was provided

Return type

qiskit.circuit.Instruction

property decompositions

Get the decompositions of the instruction from the SessionEquivalenceLibrary.

property definition

Return definition in terms of other basic gates.

property duration

Get the duration.

inverse()[source]

Return inverse RZX gate (i.e. with the negative rotation angle).

is_parameterized()

Return True .IFF. instruction is parameterized else False

property label

Return gate label

Return type

str

mirror()

DEPRECATED: use instruction.reverse_ops().

Returns

a new instruction with sub-instructions

reversed.

Return type

qiskit.circuit.Instruction

property params

return instruction params.

power(exponent)

Creates a unitary gate as gate^exponent.

Parameters

exponent (float) – Gate^exponent

Returns

To which to_matrix is self.to_matrix^exponent.

Return type

qiskit.extensions.UnitaryGate

Raises

CircuitError – If Gate is not unitary

qasm()

Return a default OpenQASM string for the instruction.

Derived instructions may override this to print in a different format (e.g. measure q -> c;).

repeat(n)

Creates an instruction with gate repeated n amount of times.

Parameters

n (int) – Number of times to repeat the instruction

Returns

Containing the definition.

Return type

qiskit.circuit.Instruction

Raises

CircuitError – If n < 1.

reverse_ops()

For a composite instruction, reverse the order of sub-instructions.

This is done by recursively reversing all sub-instructions. It does not invert any gate.

Returns

a new instruction with

sub-instructions reversed.

Return type

qiskit.circuit.Instruction

to_matrix()[source]

Return a numpy.array for the RZX gate.

property unit

Get the time unit of duration.

validate_parameter(parameter)

Gate parameters should be int, float, or ParameterExpression