This is the same protocol behind the 2019 Google Sycamore "quantum supremacy" claim, run here on a full complex statevector of dimension D = 2N built entirely from scratch (no library): each layer applies an independent random single-qubit rotation Rx/y/z(θ) to every qubit, then a fixed entangling gate (controlled-Z) across alternating neighbour bonds — a brickwork pattern identical in spirit to a real superconducting-qubit chip.
|ψ⟩ after L layers = ∏ᴸ ( CZ-layer · ⊗ᵢ Rᵢ(θᵢ) ) |0…0⟩
P(x) = |⟨x|ψ⟩|² exactly, for every one of the D = 2ᴺ basis strings x
For N ≤ 8 this exact statevector (D ≤ 256) is small enough to track every amplitude directly — no sampling, no approximation. The top chart shows the exact, sorted output probabilities of the current circuit, live.
A single sufficiently-scrambled random circuit's output looks like a Haar-random pure state: its probabilities x = p·D (rescaled so the mean is exactly 1) follow the Porter–Thomas distribution. The exact finite-dimensional law is x's marginal from a flat Dirichlet(1,…,1) split of unit probability across D outcomes — mean(x) = 1 exactly for any D, and var(x) = (D−1)/(D+1) exactly, only approaching the textbook asymptotic P(x) = e−x (mean 1, var 1) as D → ∞. One circuit's D probabilities are too few points to see the shape clearly — so "Run Batch" builds many independent random circuits at a fixed scrambling depth (deep enough that adding more layers no longer changes the pooled statistics — the circuit has reached its long-time value), pools every one of their D exact probabilities together, and histograms them. As the sample grows, the pooled histogram converges onto the theoretical curve (solid line) and the readouts above converge onto mean(x) = 1 and the exact finite-D variance — a direct numerical confirmation that random circuit sampling really does produce Porter–Thomas "speckle" statistics, the same signature real quantum-supremacy experiments are graded against.
Exact: mean(x) = 1, var(x) = (D−1)/(D+1)
Asymptotic (D→∞): P(x) = e⁻ˣ, mean = var = 1
Fit quality: max over histogram bins of |empirical density − e⁻ˣ|