RZXGate

class RZXGate(theta, label=None)

Bases: Gate

A parametric 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).

Can be applied to a QuantumCircuit with the rzx() method.

Circuit Symbol:

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

Matrix Representation:

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

নোট

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.

Methods Defined Here

inverse	Return inverse RZX gate (i.e.
power	Raise gate to a power.

Attributes

condition_bits

Get Clbits in condition.

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 instruction label

name

Return the name.

num_clbits

Return the number of clbits.

num_qubits

Return the number of qubits.

params

return instruction params.

unit

Get the time unit of duration.