Cross-Resonance Gate

The microwave-driven entangling interaction that produces CNOT on fixed-frequency superconducting qubits.

Definition

The cross-resonance (CR) gate entangles two fixed-frequency superconducting qubits by driving the control qubit at the frequency of the target. The drive induces a target rotation whose direction depends on the control's state — effectively a ZX-type interaction — which, after calibration, yields a CNOT without needing tunable qubit frequencies.

The generated interaction

To leading order the CR drive generates a term proportional to Z_control ⊗ X_target, i.e. an RZX rotation. Unwanted terms (IX, ZI, ZZ) also appear and must be suppressed. A rotary-echo or echoed-CR sequence cancels the parasitic terms, leaving a clean ZX interaction that integrates to a maximally entangling gate.

Why it matters

Fixed-frequency transmons are less sensitive to flux noise and have long coherence, but they cannot be tuned into resonance to entangle. Cross-resonance solves this by using an all-microwave drive, making it the entangling mechanism behind much of IBM's fixed-frequency hardware.

Echoed cross-resonance

The practical gate is the echoed CR: two CR pulses of opposite sign sandwiching an X on the control. The echo cancels the IX and ZZ error terms while the desired ZX term adds, dramatically improving fidelity. See RZX gate, which is the ideal interaction CR realizes.

# echoed CR structure (schematic)
# CR(+)  ; X on control ; CR(-) ; X on control
def echoed_cr_pulses():
    return ['CR+', 'Xc', 'CR-', 'Xc']

See also native gate sets for where CR fits in the superconducting basis.