This reproduces the actual statistical certificate used in Google's 2019 Sycamore "quantum supremacy" paper: linear cross-entropy benchmarking (XEB). A random circuit on N qubits defines an ideal output distribution P(x) over D = 2N bitstrings that (for a sufficiently scrambled circuit) follows the Porter–Thomas law — most bitstrings are near-equally likely, but a thin exponential tail of "lucky" strings carries most of the weight, the signature of quantum chaos. That fixed distribution is generated once per qubit count and never changes while you sample from it.
Linear XEB estimator, from M measured bitstrings x₁…x_M:
F_XEB = (D / M) · Σᵢ P(xᵢ) − 1
A perfect noiseless sampler draws exactly from P(x), which makes F_XEB converge to 1. Real hardware is noisy: with probability F it samples from P(x), and with probability (1 − F) a gate error scrambles the outcome to uniformly random — a depolarizing model — so the honest estimator converges to the true hardware fidelity F, exactly the number hardware teams report. The fidelity slider sets F directly here (in the real experiment it comes from multiplying per-gate error rates across the whole circuit).
The point of the test is that it also rejects imposters: a classical adversary that cannot afford to track the full 2N-dimensional state has to guess bitstrings some other way. A pure uniform guess pushes F_XEB to 0 in expectation — indistinguishable in this metric from noise. The adversary sophistication γ slider models a smarter classical attacker that weights its guesses by P(x)γ using partial, low-order correlations it can afford to compute classically (γ = 0 is a uniform guess, γ = 1 would require the full state and isn't classically cheap) — you can watch how much of the honest score such a shortcut can claw back, and how far it still falls short of real hardware.
Spoof draw: x ~ P(x)^γ / Σ P(x)^γ (γ=0 → uniform, γ=1 → perfect)
The 3D bars group all D outcomes into probability-rank bins (rarest → most likely, left to right) and show where each stream's samples are landing live — the honest stream should pile up on the high-probability bins on the right; a weak spoof stays flat.