Optimal Disorder Strength for Approximate Unitary k-Designs in Spin Chains
Abstract
Quantum simulators with disordered spin Hamiltonians offer a natural route to Haar-like unitary ensembles, but face a competition. Disorder enhances randomness, yet strong disorder suppresses dynamics and localizes the system in the computational basis. This suggests an optimal intermediate disorder strength for approximating a unitary k-design, which has so far been identified mainly empirically. Here, we develop a perturbative theory for the moment gap Δ(k), the deviation of the k-th moment operator from its Haar value, for the full spectrum of the interacting XXZ chain in a fixed-magnetization sector. We find the non-monotonic asymptotic scalings ln Δ(k)∼ −σ2 at weak disorder and ln Δ(k)∼ −1/σ at strong disorder, whose crossover sets an optimal disorder strength σc∼ Jxy, where Jxy is the energy scale of the non-random interaction. We further show that a multi-quench protocol further suppresses the gap exponentially in the number of quenches. Numerical simulations confirm both scalings and this exponential suppression, and we demonstrate the usefulness of the protocol for classical shadow tomography. This work suggests that strong disorder may block both thermalization and the formation of unitary designs through a common localization mechanism.
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BibTeX
@misc{zhou2026optimal,
title = {{Optimal Disorder Strength for Approximate Unitary $k$-Designs in Spin Chains}},
author = {Tian-Gang Zhou and Yi-Neng Zhou and Thierry Giamarchi and Julian Sonner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18954},
eprint = {2610.18954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18954},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Disorder Strength for Approximate Unitary k-Designs in Spin Chains
AU - Tian-Gang Zhou
AU - Yi-Neng Zhou
AU - Thierry Giamarchi
AU - Julian Sonner
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18954
UR - https://arxiv.org/abs/2610.18954
AB - Quantum simulators with disordered spin Hamiltonians offer a natural route to Haar-like unitary ensembles, but face a competition. Disorder enhances randomness, yet strong disorder suppresses dynamics and localizes the system in the computational basis. This suggests an optimal intermediate disorder strength for approximating a unitary k-design, which has so far been identified mainly empirically. Here, we develop a perturbative theory for the moment gap Δ^(k), the deviation of the k-th moment operator from its Haar value, for the full spectrum of the interacting XXZ chain in a fixed-magnetization sector. We find the non-monotonic asymptotic scalings ln Δ^(k)∼ −σ^2 at weak disorder and ln Δ^(k)∼ −1/σ at strong disorder, whose crossover sets an optimal disorder strength σ_c∼ J_xy, where J_xy is the energy scale of the non-random interaction. We further show that a multi-quench protocol further suppresses the gap exponentially in the number of quenches. Numerical simulations confirm both scalings and this exponential suppression, and we demonstrate the usefulness of the protocol for classical shadow tomography. This work suggests that strong disorder may block both thermalization and the formation of unitary designs through a common localization mechanism.
ER -