发表机构
Stanford University; University of Michigan（斯坦福大学; 密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明一维非对易局域自旋系统在任意正温度下量子吉布斯采样器快速混合，改进收敛时间至对数阶，并揭示谱隙低估平衡速度的机理。
AI 中文摘要
我们证明了一般一维非对易局域自旋系统在任意固定正温度下量子吉布斯采样器的快速混合。特别地，对于包含 $n$ 个格点的系统，我们表明文献[CKG2023]中的Kubo-Martin-Schwinger（KMS）细致平衡吉布斯采样器从任意初始态出发，在时间 $O(\log^4(n)+\log(1/\epsilon))$ 内收敛到迹范数误差 $\epsilon$。这改进了先前的 $O(n+\log(1/\epsilon))$ 界，并确立了超越高温和弱相互作用区间的快速混合。我们的证明解释了为何由谱隙捕获的最慢衰减率会显著低估平衡速度。关键洞察在于，远离平衡的态必然在低权重泡利算子张成空间之外具有可观分量，而那里适用更强的耗散估计。这产生两个弛豫阶段：从不良初始态的双指数收敛，随后是接近平衡时的指数收敛。我们将这一机制转化为一个通过超越谱隙的谱分析来证明快速混合的框架。我们的衰减估计等价于夹心Rényi-2散度下的指数收敛。结合谱隙，它还蕴含速率为 $\Omega(\log^{-4}(n))$ 的修正对数Sobolev不等式。
英文摘要
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 $Ω(\log^{-4}(n))$.
Comments38 pages, 1 figure
