RZXGate

class RZXGate(theta)

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.

Attributes

RZXGate.decompositions	Get the decompositions of the instruction from the SessionEquivalenceLibrary.
RZXGate.definition	Return definition in terms of other basic gates.
RZXGate.label	Return gate label
RZXGate.params	return instruction params.

Methods

RZXGate.add_decomposition(decomposition)	Add a decomposition of the instruction to the SessionEquivalenceLibrary.
RZXGate.assemble()	Assemble a QasmQobjInstruction
RZXGate.broadcast_arguments(qargs, cargs)	Validation and handling of the arguments and its relationship.
RZXGate.c_if(classical, val)	Add classical condition on register classical and value val.
RZXGate.control([num_ctrl_qubits, label, …])	Return controlled version of gate.
RZXGate.copy([name])	Copy of the instruction.
RZXGate.inverse()	Return inverse RZX gate (i.e.
RZXGate.is_parameterized()	Return True .IFF.
RZXGate.mirror()	For a composite instruction, reverse the order of sub-gates.
RZXGate.power(exponent)	Creates a unitary gate as gate^exponent.
RZXGate.qasm()	Return a default OpenQASM string for the instruction.
RZXGate.repeat(n)	Creates an instruction with gate repeated n amount of times.
RZXGate.to_matrix()	Return a Numpy.array for the gate unitary matrix.