Random unitary circuits with constant spectral gap
Abstract
We prove constant lower bounds for the spectral gap of the following random walks on unitary groups SU(2n) on n qubits. (i) Random Pauli Rotation: choose an n-qubit Pauli operator P and an angle θ∈ [0,2π), both uniformly at random, and apply ei θP. (ii) Brickwork Random Unitary Circuit: choose n−1 unitaries Ui uniformly at random from SU(4) (including central phases of the parent group) independently, and apply U2j−1 on two qubits $2j-1, 2j$ and then U2j on two qubits 2j, 2j+1. Importantly, the gap bounds are independent of n and apply for all finite dimensional unitary representations of SU(2n) uniformly. Our results immediately imply that the Random Pauli Rotation and Brickwork Random Unitary Circuit converge to approximate unitary t-designs after 𝒪(nt) steps. As a consequence, we improve implications for the complexity growth of random unitary circuits as a function of the depth. The existing argument can be applied almost verbatim, but previously the depth had to be 𝒪(2cn) for some c < 1. Our unconditional constant spectral gap removes this restriction. We prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit. The former implies convergence to approximate orthogonal t-designs. In this case, the implied mixing time with respect to a constant additive and multiplicative error is optimal (up to an absolute constant factor) for t=2.
Related
- Spectral gaps and slow modes of Pauli rotations and random quantum circuits
- Finite-bath projected ensembles in dual-unitary circuits
- Averages of traces and ratios of characteristic polynomials of random unitary matrices
- Edgeworth expansions of extreme eigenvalue distributions for random unitary ensembles
- The Lq-bounds for Derivatives of Unitary Developments of Random Continuous Geometric Rough Paths
BibTeX
@misc{baer2026random,
title = {{Random unitary circuits with constant spectral gap}},
author = {Tim Baer and Jeongwan Haah},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.20919},
eprint = {2607.20919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.20919},
}
RIS
TY - UNPB
TI - Random unitary circuits with constant spectral gap
AU - Tim Baer
AU - Jeongwan Haah
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.20919
UR - https://arxiv.org/abs/2607.20919
AB - We prove constant lower bounds for the spectral gap of the following random walks on unitary groups SU(2^n) on n qubits. (i) Random Pauli Rotation: choose an n-qubit Pauli operator P and an angle θ∈ [0,2π), both uniformly at random, and apply e^i θP. (ii) Brickwork Random Unitary Circuit: choose n−1 unitaries U_i uniformly at random from SU(4) (including central phases of the parent group) independently, and apply U_2j−1 on two qubits $2j-1, 2j$ and then U_2j on two qubits 2j, 2j+1. Importantly, the gap bounds are independent of n and apply for all finite dimensional unitary representations of SU(2^n) uniformly. Our results immediately imply that the Random Pauli Rotation and Brickwork Random Unitary Circuit converge to approximate unitary t-designs after 𝒪(nt) steps. As a consequence, we improve implications for the complexity growth of random unitary circuits as a function of the depth. The existing argument can be applied almost verbatim, but previously the depth had to be 𝒪(2^cn) for some c < 1. Our unconditional constant spectral gap removes this restriction. We prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit. The former implies convergence to approximate orthogonal t-designs. In this case, the implied mixing time with respect to a constant additive and multiplicative error is optimal (up to an absolute constant factor) for t=2.
ER -