Ideal random quantum circuits pass the LXEB test

TL;DR

This paper demonstrates that noiseless random quantum circuits, especially those forming unitary designs at certain depths, pass the LXEB test with high probability, revealing properties of their output distributions.

Contribution

The paper provides rigorous analysis showing that random quantum circuits of specific depths pass the LXEB test, connecting circuit depth, unitary design properties, and output distribution concentration.

Findings

Linear depth circuits pass LXEB with probability 1 - O(1/√k)

Depth ~ n^2 circuits pass LXEB with probability 1 - O(e^{-k log(n)/n})

Output probabilities exhibit Porter-Thomas distribution at near-quadratic depths

Abstract

We show that noiseless random quantum circuits pass the linear cross-entropy benchmark (LXEB) test with high probability. If the circuits are linear depth, and thus form unitary 4-designs, the LXEB test is passed with probability $1-O(1/\sqrt{k})$, where $k$ is the number of independently drawn samples from the output distribution of the random circuit. If the circuits are of depth $\tilde O(n^2)$, and thus form unitary $n$-designs, the LXEB test is passed with probability $1-O(e^{-k \log(n)/n})$. In proving our results, we show strong concentration of the random circuit collision probability at linear depth and establish that the tails of the distribution of random circuit output probabilities start to resemble Porter-Thomas at near-quadratic depths. Our analysis employs higher moments and high-degree approximate designs.