Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension
Abstract
We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of n sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error ε in time O(log4(n)+log(1/ε)). This improves the previous O(n+log(1/ε)) bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate Ω(log−4(n)).
Related
- Rapid mixing for Gibbs samplers for quantum spin systems with all-to-all interactions
- Second-order perturbation bounds for Gibbs samplers under strong spatial mixing
- Rényi mixing bounds for kinetic samplers via L2 hypocoercivity
- A priori regularity of the reverse heat flow and dimension-dependent complexity of higher-order diffusion samplers
- Hypercontractivity and entropy decay for infinite-dimensional quantum Gibbs samplers
BibTeX
@misc{chen2026spectral,
title = {{Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension}},
author = {Hongrui Chen and Zhiyan Ding},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05848},
eprint = {2610.05848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05848},
}
RIS
TY - UNPB
TI - Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension
AU - Hongrui Chen
AU - Zhiyan Ding
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.05848
UR - https://arxiv.org/abs/2610.05848
AB - We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of n sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error ε in time O(log^4(n)+log(1/ε)). This improves the previous O(n+log(1/ε)) bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate Ω(log^−4(n)).
ER -