Cross-entropy benchmarking fidelity vs circuit depth and qubit count

Linear cross-entropy benchmarking (XEB) fidelity from random circuit sampling, swept over circuit depth (number of cycles) and the number of qubits in the circuit – the system-level benchmark used where per-gate RB no longer scales, once a circuit is too large to decompose into a handful of independently characterized gates. Each cycle applies one layer of random single-qubit gates and one layer of two-qubit entangling gates across the whole register, so a fixed per-cycle error per qubit pair compounds once per cycle and once per pair – fidelity falls off exponentially in the product of depth and qubit count, not in either alone, which is what makes both axes matter simultaneously here rather than one being swept just to check the other still works.

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import numpy as np
import polars as pl
import plotpress

ERROR_PER_CYCLE_PAIR = 0.0065      # per-cycle, per-qubit-pair error contribution
rng = np.random.default_rng(1708)

depth = np.arange(1, 41)               # circuit cycles
n_qubits = np.arange(2, 26)             # qubits in the random circuit
D, NQ = np.meshgrid(depth, n_qubits)

pairs_per_cycle = np.floor(NQ / 2.0)
xeb_fidelity = (1.0 - ERROR_PER_CYCLE_PAIR) ** (D * pairs_per_cycle)
xeb_fidelity *= 1.0 + rng.normal(0.0, 0.03, xeb_fidelity.shape)
xeb_fidelity = np.clip(xeb_fidelity, 1e-4, 1.0)

# One row per swept (depth, qubit count) circuit -- sorted before the
# reshape below so the pivot back to a grid is correct regardless of order.
sweep = pl.DataFrame({
    "depth": D.ravel(),
    "n_qubits": NQ.ravel(),
    "log_fidelity": np.log10(xeb_fidelity).ravel(),
}).sort(["n_qubits", "depth"])

depth_axis = sweep["depth"].unique().sort().to_numpy()
n_qubits_axis = sweep["n_qubits"].unique().sort().to_numpy()
log_fidelity = sweep["log_fidelity"].to_numpy().reshape(n_qubits_axis.size, depth_axis.size)

fig, ax = plotpress.subplots(figsize=(7.6, 5.4))
mesh = ax.pcolormesh(depth_axis, n_qubits_axis, log_fidelity, cmap="viridis")
bar = fig.colorbar(mesh, ax=ax)
bar.set_title("log10 F_XEB")
ax.set_xlabel("circuit depth (cycles)")
ax.set_ylabel("number of qubits")
ax.set_title("XEB fidelity falls off in depth times qubit count, not either alone")
fig.tight_layout()