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.

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. 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. Return inverse RZX gate (i.e. Return True .IFF. For a composite instruction, reverse the order of sub-gates. RZXGate.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. Return a Numpy.array for the gate unitary matrix.