Local random quantum circuits converge to the Porter-Thomas distribution in polynomial depth
Abstract
Porter-Thomas statistics are a characteristic feature of the output distribution of random quantum states and, more broadly, chaotic quantum many-body systems. Convergence to Porter-Thomas plays a central role in random circuit sampling and experimental demonstrations of quantum advantage, where the output statistics of low-depth random quantum circuits are expected to be approximately Porter-Thomas, despite the absence of a rigorous proof of convergence. We show that the output distribution of polynomial-depth brickwork random circuits converges inverse-polynomially in total variation distance to the Porter-Thomas distribution. Specifically, consider the output probability distribution over a fixed bitstring of a local random quantum circuit, constructed from nearest-neighbor Haar random gates. Then, for any m ≥ 0, the distribution corresponding to circuits of depth O(n2m+1log(n)) is at most O(1/nm) far in total variation distance from the Porter-Thomas distribution. Our proof uses moment bounds from approximate designs, analytic estimates for characteristic functions, and a local anticoncentration property for inverse moments.
Related
- Polynomial-time additive-error estimation of output probabilities for shallow quantum circuits
- A polynomial-time classical sampler for noisy quantum circuits from statistical mechanics
- Random quantum circuits, chaos and quantum thermalization
- Rare-History Transitions in Temporally Random Integrable Quantum Circuits
- One Gate at a Time: Complexity Growth in Random Quantum Circuits
BibTeX
@misc{sen2026local,
title = {{Local random quantum circuits converge to the Porter-Thomas distribution in polynomial depth}},
author = {Aniruddha Sen and Nicholas Hunter-Jones},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02125},
eprint = {2610.02125},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02125},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local random quantum circuits converge to the Porter-Thomas distribution in polynomial depth
AU - Aniruddha Sen
AU - Nicholas Hunter-Jones
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02125
UR - https://arxiv.org/abs/2610.02125
AB - Porter-Thomas statistics are a characteristic feature of the output distribution of random quantum states and, more broadly, chaotic quantum many-body systems. Convergence to Porter-Thomas plays a central role in random circuit sampling and experimental demonstrations of quantum advantage, where the output statistics of low-depth random quantum circuits are expected to be approximately Porter-Thomas, despite the absence of a rigorous proof of convergence. We show that the output distribution of polynomial-depth brickwork random circuits converges inverse-polynomially in total variation distance to the Porter-Thomas distribution. Specifically, consider the output probability distribution over a fixed bitstring of a local random quantum circuit, constructed from nearest-neighbor Haar random gates. Then, for any m ≥ 0, the distribution corresponding to circuits of depth O(n^2m+1log(n)) is at most O(1/n^m) far in total variation distance from the Porter-Thomas distribution. Our proof uses moment bounds from approximate designs, analytic estimates for characteristic functions, and a local anticoncentration property for inverse moments.
ER -