Cross-Resonance Gate
How the ZX Interaction Builds an Entangling Operation
The cross-resonance technique solves a specific hardware problem. Fixed-frequency transmons enjoy long coherence times precisely because they have no flux-tunable junction loop to couple in flux noise, but that same rigidity removes the obvious way to bring two qubits into resonance for an exchange interaction. Cross-resonance sidesteps the issue entirely: instead of moving qubit frequencies, the control qubit is driven with a microwave pulse whose carrier sits at the target qubit's 0-to-1 transition frequency. The static capacitive or bus-mediated coupling between the two qubits then mixes that drive down into a target-frame rotation.
When the resulting Hamiltonian is expanded in the two-qubit Pauli basis, the term that does useful work is ZX, meaning the target precesses about its X axis with a sign set by whether the control sits in its ground or excited state. That conditional rotation is the entangling ingredient. Accompanying the wanted ZX term are several spectator terms: IX from direct classical crosstalk of the drive onto the target, ZI as a Stark-type shift of the control, and an always-on ZZ from the dispersive coupling. Gate calibration is largely the art of maximizing ZX while driving these parasitic terms toward zero.
The drive amplitude cannot be raised without limit. Because the transmon is only weakly anharmonic, an aggressive pulse leaks population into the second-excited state and into neighboring qubits. The qubit-qubit detuning therefore has to stay larger than the drive amplitude yet smaller than the transmon anharmonicity, which puts the practical detuning window between roughly 50 and 200 MHz for anharmonicities near 320 MHz.
Echoed Pulses and Active Cancellation
A single bare pulse leaves too much IX and ZI contamination to reach high fidelity, so production gates use an echoed cross-resonance sequence. The CR pulse is split into two equal halves of opposite amplitude with a control-qubit X pi pulse inserted between them. The wanted ZX term flips sign with the control and so accumulates coherently, while the static IX and ZI errors flip sign without the control and cancel. A weak compensation tone applied directly to the target, often called a rotary or target-cancellation drive, mops up the residual IX. With these refinements, randomized-benchmarking error rates of 0.5 to 1 percent are routine on modern devices.
Governing Relations
ΩZX ≈ (J × Ω) / Δ × [α / (α + Δ)]
CNOT-equivalent pulse area:
θZX = ΩZX × tg = π/2
Approximate gate time:
tg ≈ (π/2) / ΩZX ≈ (π × Δ) / (2 × J × Ω)
Where J = qubit-qubit coupling (1 to 4 MHz), Ω = drive amplitude on the control (10 to 60 MHz), Δ = control-target detuning (50 to 200 MHz), and α = transmon anharmonicity (≈ −320 MHz). Example: J = 3 MHz, Ω = 30 MHz, Δ = 100 MHz → ΩZX ≈ 0.7 MHz, giving tg ≈ 350 ns.
Two-Qubit Gate Family Comparison
| Gate | Mechanism | Qubit Type | Typical Time | Reported Error | Control Hardware |
|---|---|---|---|---|---|
| Cross-Resonance | Microwave drive at neighbor frequency, ZX | Fixed-frequency transmon | 150 to 400 ns | 0.5 to 1% | All-microwave |
| Tunable CZ (flux) | Flux pulse into 11-02 avoided crossing | Flux-tunable transmon | 30 to 60 ns | 0.1 to 0.5% | Fast flux + microwave |
| iSWAP / parametric | Modulated coupler at sum/difference freq | Tunable coupler | 40 to 100 ns | 0.2 to 0.6% | RF coupler drive |
| Molmer-Sorensen | Spin-dependent force via shared mode | Trapped ion | 10 to 100 μs | 0.1 to 0.5% | Laser / RF |
Frequently Asked Questions
How does the cross-resonance gate generate a CNOT without tuning either qubit?
The control transmon is driven at the target qubit's transition frequency. Through the fixed bus coupling (J ≈ 1 to 4 MHz), this produces a dominant ZX term, a target X rotation whose sign tracks the control state. Calibrating that to a π/2 ZX rotation and adding single-qubit Z and X frames is equivalent to a CNOT. No flux line or frequency excursion is required, which is why fixed-frequency transmons rely on it.
Why is an echoed cross-resonance pulse sequence used?
A bare drive also produces IX, ZI, and ZZ error terms from crosstalk and dispersive coupling. The echoed sequence uses two CR halves of opposite sign with a control X π pulse between them: the wanted ZX adds while static IX and ZI cancel. A rotary cancellation tone on the target removes residual IX. Together these reach 0.5 to 1 percent two-qubit error.
What detuning between control and target qubits gives the fastest gate?
The ZX rate scales as J × Ω / Δ, while leakage and frequency collisions worsen as Δ shrinks. The sweet spot keeps Δ near 50 to 200 MHz, larger than the drive Ω but smaller than the anharmonicity (≈ 320 MHz), giving 150 to 400 ns gates. Collision points at Δ = 0, Δ = α, and Δ = α/2 must be avoided in frequency planning.