quantum gate directory

The $zx$ rotation is generated by the two-body interaction $Z \otimes X$:

$$ R_{zx}(\theta) = \exp\left(-i \frac{\theta}{2} Z \otimes X\right) $$

In the computational basis:

$$ \begin{bmatrix} \cos\tfrac{\theta}{2} & -i\sin\tfrac{\theta}{2} & 0 & 0 \\ -i\sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} & 0 & 0 \\ 0 & 0 & \cos\tfrac{\theta}{2} & i\sin\tfrac{\theta}{2} \\ 0 & 0 & i\sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} \end{bmatrix} $$

The block structure makes the action clear: the target is rotated by $R_x(\theta)$ when the first qubit is $|0\rangle$ and by $R_x(-\theta)$ when it is $|1\rangle$.

Properties

• Unlike its cousins $R_{xx}$, $R_{yy}$, and $R_{zz}$, the generator is asymmetric, so the two qubits play different roles.
• Related to $R_{zz}$ by a basis change on the target: $R_{zx}(\theta) = (I \otimes H) \, R_{zz}(\theta) \, (I \otimes H)$.
• A single $R_{zx}(\pi/2)$ plus one-qubit gates yields a CNOT.

Usage

• The cross-resonance interaction is what an all-microwave two-qubit gate on fixed-frequency transmons actually implements; IBM's native ECR gate is two echoed $R_{zx}(\pi/4)$ half-pulses with an $X$ on the control in between, which cancels the always-on error terms.
• Direct $R_{zx}(\theta)$ pulses enable shorter circuits for quantum simulation than compiling through CNOTs.