Switch an interaction off. Then use it for a gate.
Two superconducting qubits never fully stop talking: a residual ZZ makes one qubit's frequency depend on the other's state. A tunable coupler cancels it at one frequency, the idle point; a flux pulse turns it up for a controlled-Z (CZ) gate. Every curve here is computed by the tested Python models (qll/hardware/tunable_coupler.py, cz_gate.py, qll/circuits/zz_ramsey.py) and written to site_data.json.
|static ZZ| vs coupler frequency · exact diagonalisation of qubit–coupler–qubit
at this coupler frequency
Idle point from the model: — GHz. Qubits at —, α = −200 MHz, gqc = 100 MHz.
conditional Ramsey on qubit 0 · neighbour in |0⟩ (blue) and |1⟩ (red) · the fringe frequencies differ by exactly ZZ
P₁(τ) = ½ − ½e−τ/T₂cos(2πfsτ), with f0 = δ + ZZ/2 and f1 = δ − ZZ/2 (virtual detuning δ = 0.5 MHz, T₂ = 40 µs). This is experiment E16: the same circuits run on a free cloud processor.
a CZ gate, simulated with three levels per transmon · qubit 1 tuned from 6.00 to 5.25 GHz and back
flux pulse ω₁(t)
|11⟩ and |20⟩ levels · they avoid each other by 2√2 g
where the |11⟩ population is
conditional phase · the running value is −2π∫ZZ dt, the final one the full simulation · a CZ needs π
ZZ right now: — MHz. Near the crossing it is about 11 MHz, so π takes a few tens of nanoseconds; with qubit 1 back at 6.00 GHz it falls to a few hundred kHz, which a tunable coupler parked at its idle point (above) would remove. Single-qubit phases are removed in software.
why the shape matters
Shaped smoothly in the |11⟩–|20⟩ mixing angle (after Martinis and Geller), the pulse slows down exactly where the two levels are closest, and the population swinging into |20⟩ all comes back: — left behind. With the same 15 ns ramps shaped linearly in frequency and recalibrated to π, — is stranded in |20⟩: leakage, the error that fast adiabatic pulses are designed to avoid.
Hybridization is not leakage: at the interaction point the dressed |11⟩ is 17 % bare |20⟩, and the bar returns to zero when the pulse ends. What counts is what is left at the end. Idealized: coherent evolution only, no T₁ or T₂.