ECR (echoed cross-resonance) gate

Definition (Qiskit convention)

$$\mathrm{ECR} = \frac{1}{\sqrt2}\begin{pmatrix} 0&0&1&i\\0&0&i&1\\1&-i&0&0\\-i&1&0&0 \end{pmatrix}$$

using Qiskit's ECRGate convention, with qubit 0 as the "control" side of the underlying RZX pulses. This matrix is maximally entangling and locally equivalent to $\mathrm{CX}$ (same Weyl-chamber point), but is not identical to it.

Action on basis states

$$\mathrm{ECR}|00\rangle = \tfrac{1}{\sqrt2}(|10\rangle - i|11\rangle), \qquad \mathrm{ECR}|10\rangle = \tfrac{1}{\sqrt2}(|00\rangle + i|01\rangle)$$

Every computational-basis input is mapped to an equal-weight superposition of two basis states on the other qubit's block, confirming maximal entangling power from a product-state input.

Decomposition

$$\mathrm{ECR} = \mathrm{RZX}(-\pi/4)\cdot (X\otimes I) \cdot \mathrm{RZX}(\pi/4)$$

$\mathrm{RZX}(\theta) = e^{-i\theta (Z\otimes X)/2}$ is itself a native two-qubit rotation driven directly by the cross-resonance microwave tone — it is not decomposed further here, since on IBM hardware it is a calibrated pulse, not a compiled circuit. Multiplying the three $4\times4$ matrices in circuit order (RZX(π/4), then an X flip on qubit 0, then RZX(-π/4)) reproduces the ECR matrix above exactly, entry by entry.

Key identities

• $\mathrm{ECR}$ is a Clifford gate per Qiskit's ECRGate documentation, since it is locally equivalent to $\mathrm{CX}$; standard Clifford-simulator and transpiler passes treat it as such.
• The "echo" structure — $\mathrm{RZX}(-\pi/4)$ and $\mathrm{RZX}(\pi/4)$ with an intervening $X$ on the control — is the standard technique (Sheldon et al. 2016) for canceling static $ZZ$ and $IX$/$IY$ crosstalk terms that a bare cross-resonance pulse would otherwise leave in the effective Hamiltonian.
• $\mathrm{RZX}$ is not part of OpenQASM 3's core stdgates.inc; it is exposed as a hardware-native gate by IBM's backend dialect, the same status ECR itself has.
• Because ECR is locally equivalent to $\mathrm{CX}$, any circuit expressed with $\mathrm{CX}$ plus single-qubit gates can be retargeted to ECR plus single-qubit gates by a transpiler, which is exactly how Qiskit compiles CX-based circuits for ECR-native IBM backends.