CZ Gate
Conditional Phase and the Adiabatic Flux Pulse
The CZ gate belongs to the controlled-phase family. Its 4x4 unitary is diagonal in the computational basis, multiplying only the |11⟩ amplitude by eiπ = -1. That single conditional sign is enough to entangle two qubits: applied to (|0⟩+|1⟩)/√2 on each line it produces a Bell-class state. What makes the gate native to superconducting hardware is that this exact phase appears for free when two transmons are momentarily tuned into a specific resonance condition. The negative anharmonicity η of a transmon places the |02⟩ level just below twice the qubit frequency, so as one qubit is detuned downward by a flux pulse the |11⟩ and |02⟩ states cross. Exchange coupling g hybridizes them with strength √2·g, opening an avoided crossing of 2√2·g, commonly 20 to 60 MHz for g/2π in the 5 to 20 MHz range.
During the pulse the |11⟩ amplitude follows an adiabatic excursion toward that crossing and back, while |01⟩ and |10⟩ have no near-resonant partner and barely move. The differential dynamical phase φ is the time integral of the energy shift of |11⟩. The pulse area, set by how close the trajectory approaches the crossing and for how long, is calibrated so φ = π. A residual single-qubit phase on each line is removed afterward with virtual-Z frame updates, leaving an ideal CZ. The hard constraint is that the excursion must be adiabatic enough to suppress real population transfer to |02⟩ (leakage) yet fast enough that decoherence over the gate window stays small.
Governing Relations
UCZ = diag(1, 1, 1, -1) = diag(1, 1, 1, eiπ)
11 to 02 avoided-crossing gap:
Δgap = 2√2 × g (g = bare exchange coupling)
Accumulated conditional phase:
φ11 = ∫ [E11(t) − E01(t) − E10(t) + E00] dt / ℏ ≡ π
Adiabaticity / leakage bound:
Pleak ≈ (∂ω/∂t)2 / Δgap4 → slow ramp near crossing
Where η = transmon anharmonicity (≈ −200 to −330 MHz), ℏ = reduced Planck constant. Example: g/2π ≈ 15 MHz → Δgap/2π ≈ 42 MHz, supporting a ≈ 40 ns CZ.
Two-Qubit Gate Comparison
| Gate | Physical mechanism | Qubit type | Typical time | Reported fidelity | Key limitation |
|---|---|---|---|---|---|
| CZ (flux) | 11/02 avoided crossing via flux pulse | Tunable transmon | 30 to 60 ns | 99.5 to 99.9% | Flux-noise dephasing, leakage |
| CNOT | CZ + two target-qubit Hadamards | Any (compiled) | ~CZ + 2 SQ | Follows CZ | Inherits CZ errors |
| Cross-resonance | Drive control at target frequency | Fixed-frequency transmon | 160 to 400 ns | 99.0 to 99.8% | Slow; classical crosstalk |
| iSWAP | 01/10 resonant exchange | Tunable transmon | 20 to 50 ns | 99.0 to 99.7% | Excitation exchange errors |
| Parametric CZ | RF-driven tunable coupler modulation | Coupler-mediated | 50 to 150 ns | 99.0 to 99.5% | Drive-induced shifts |
Frequently Asked Questions
How does a flux-pulse CZ gate use the 11 to 02 avoided crossing?
A fast flux pulse lowers one transmon's frequency so |11⟩ approaches the non-computational |02⟩ level. Anharmonicity lets the two states hybridize with strength √2·g, opening a gap of 2√2·g (20 to 60 MHz). The |11⟩ amplitude makes an adiabatic excursion toward the crossing and back, accumulating extra phase relative to |01⟩ and |10⟩. Setting the pulse area so that differential phase reaches π yields an exact CZ in 30 to 60 ns.
What limits CZ gate fidelity in superconducting qubits?
Leakage and decoherence dominate. Net-zero and Slepian flux-pulse shapes hold residual |02⟩ population below 0.001 per gate. Decoherence error scales as tgate/T2, so a 40 ns gate against a 30 µs T2 adds 0.001 to 0.002. Flux noise, flux-line settling (corrected by pre-distortion), and residual ZZ coupling add the rest. Interleaved randomized benchmarking reports 99.5 to 99.9 percent.
How does a CZ gate differ from a CNOT and a cross-resonance gate?
CZ and CNOT are equivalent up to single-qubit rotations: a CZ between two target-qubit Hadamards is a CNOT, so compilers treat CZ as the native entangler. Physically, a CZ uses fast flux tuning on tunable transmons (sub-60 ns), while a cross-resonance gate drives the control at the target's frequency on fixed-frequency qubits, needs no flux line, but runs 160 to 400 ns. The trade is speed versus flux-noise dephasing.