Cross-Entropy Benchmarking: Lorenz Curve

Circuit

Qubits (grid) 9

Hardware fidelity F 85%

Adversary sophistication γ 0.00

Sampling rate 200/s

Live Readout

Hilbert dim D = 2^N

512

Samples drawn

Honest F_XEB estimate

0.000

Spoof F_XEB estimate

0.000

True hardware fidelity

0.850

Std. error (honest)

±0.000

How it works

Same statistical certificate as Google's 2019 Sycamore "quantum supremacy" paper — linear cross-entropy benchmarking (XEB) — but drawn as flat 2D statistics instead of a 3D scene. A random circuit's ideal output distribution P(x) over D = 2^N bitstrings follows the Porter–Thomas law: a thin exponential tail of "lucky" strings carries most of the probability mass, the fingerprint of quantum chaos.

The top plot is the distribution's Lorenz curve: sort all D outcomes ascending by probability, plot cumulative probability mass against cumulative outcome fraction. A uniform distribution traces the diagonal; Porter–Thomas bows sharply below it then rockets up at the right edge — the top-ranked few percent of outcomes hold a hugely disproportionate share of the mass. Live sample dots and a rank histogram show exactly where each stream's draws are landing on that curve.

Linear XEB estimator, from M measured bitstrings x₁…x_M:
F_XEB = (D / M) · Σᵢ P(xᵢ) − 1

The bottom plot is the running XEB estimate itself vs. sample count. A perfect noiseless sampler converges to 1; real hardware samples from P(x) with probability F and uniformly at random otherwise (depolarizing noise), so the honest curve converges to the true hardware fidelity F — the fidelity slider sets it directly. A classical adversary that can't track the full 2^N-dimensional state has to guess some other way; the γ slider models one that weights guesses by P(x)^γ (γ=0 uniform, γ=1 would need the full state). Watch how much score such a shortcut claws back, and how far short of real hardware it still falls.

Spoof draw:  x ~ P(x)^γ / Σ P(x)^γ    (γ=0 → uniform, γ=1 → perfect)

Honest quantum sampler Classical spoof Diagonal = uniform reference