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arXiv:2610.05848

Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension

Hongrui Chen, Zhiyan Ding

quant-phmath-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations• Revised Oct 21, 2026 (v2)
v1 · Oct 5, 2026v2 · Oct 21, 2026
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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 $O(\log^4(n)+\log(1/ε))$. 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 $Ω(\log^{-4}(n))$.

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38 pages, 1 figure

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