CZ gate calibration: the 11-02 avoided crossing chevron

Population transferred from |1,1> into the leakage level |0,2>, swept over the detuning of a tunable coupler and the interaction time – the calibration map for a controlled-phase gate built on this avoided crossing between two-qubit levels, rather than a single qubit’s own drive detuning (compare Qubit Rabi chevron). Bringing |1,1> and |0,2> into resonance and back once around this crossing accumulates the conditional phase a CZ gate needs; the coupler detuning and gate time that do that with the least residual population left behind in |0,2> – reading as the first chevron fringe closing back down to zero exactly at delta = 0 – are the two numbers this measurement exists to extract.

The math is the same driven-two-level form as every other chevron in this gallery, because it genuinely is the same physics – a two-level system brought through an avoided crossing – applied to a different pair of levels for a different purpose.

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import numpy as np
import polars as pl
import plotpress

G_MHZ = 10.0                 # 11-02 coupling strength
T2_NS = 400.0                  # leakage-level-limited coherence time
rng = np.random.default_rng(303)

detuning = np.linspace(-60.0, 60.0, 320)      # MHz, coupler detuning to |02>
duration = np.linspace(0.0, 60.0, 280)         # ns
D, T = np.meshgrid(detuning, duration)

g = 2.0 * np.pi * G_MHZ * 1e-3               # rad/ns
delta = 2.0 * np.pi * D * 1e-3                # rad/ns
omega_eff = np.hypot(2 * g, delta)

contrast = np.exp(-T / T2_NS)
p_02 = contrast * (4 * g ** 2 / omega_eff ** 2) * np.sin(omega_eff * T / 2.0) ** 2
p_02 += rng.normal(0.0, 0.012, p_02.shape)

# One row per swept (detuning, duration) shot -- sorted before the reshape
# below so the pivot back to a grid is correct regardless of row order.
sweep = pl.DataFrame({
    "detuning_mhz": D.ravel(),
    "duration_ns": T.ravel(),
    "p_02": p_02.ravel(),
}).sort(["duration_ns", "detuning_mhz"])

detuning_axis = sweep["detuning_mhz"].unique().sort().to_numpy()
duration_axis = sweep["duration_ns"].unique().sort().to_numpy()
p_02 = sweep["p_02"].to_numpy().reshape(duration_axis.size, detuning_axis.size)

fig, ax = plotpress.subplots(figsize=(7.6, 5.2))
mesh = ax.pcolormesh(detuning_axis, duration_axis, p_02, cmap="viridis", vmin=0.0, vmax=1.0)
bar = fig.colorbar(mesh, ax=ax)
bar.set_title("P(02)")
ax.set_xlabel("coupler detuning to |02> (MHz)")
ax.set_ylabel("interaction time (ns)")
ax.set_title(f"CZ 11-02 avoided crossing, coupling g = {G_MHZ:.0f} MHz")
fig.tight_layout()