发表机构
Inria, Télécom Paris, LTCI, Institut Polytechnique de Paris; Department of Mathematical Sciences, University of Copenhagen（法国国家信息与自动化研究所，巴黎高等电信学校，信息与通信学院，巴黎理工学院; 哥本哈根大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过量子Dobrushin--Shlosman条件将经典快速混合理论推广至非交换量子系统，提出平滑热浴动力学，证明有限温度自旋链的对数迹范数混合及稳定性，并给出制备Gibbs态的量子算法。
AI 中文摘要
经典Dobrushin--Shlosman理论使用热浴块更新来识别建立Gibbs采样器快速混合的尖锐温度阈值。特别是，这些混合保证即使在更严格的单点条件失效时也常常成立。我们通过一个用于有限块动力学的量子Dobrushin--Shlosman条件（我们称之为平滑热浴动力学）将此方法扩展到非交换量子晶格系统。我们在两种情况下建立该条件。对于每个固定有限温度下的有限范围量子自旋链，我们证明了系统规模的对数迹范数混合，归一化界限在点场上均匀，并在逆温度上单指数。我们还证明了在多项式体积增长的图上，相互作用（可能非交换）参考哈密顿量在小局部扰动下的稳定性。对于具有有界局部相互作用的经典参考，均匀强空间混合蕴含我们的量子条件。在有界局部项和固定的物理、几何和块参数下，所得采样器使用$N\operatorname{polylog}(N/\varepsilon)$个门和经典操作制备Gibbs态至迹范数误差$\varepsilon$。我们的块更新将局部Gibbs重置与量子信念传播相结合，以纳入跨块边界的相互作用，产生精确保持Gibbs的KMS可逆通道。两个独立可调参数控制收缩：内部演化时间抑制块内点的影响，而块大小允许内部收缩主导边界影响。这种有限时间弛豫在经典热浴块更新中没有对应物。
英文摘要
Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error $\varepsilon$ using $N\operatorname{polylog}(N/\varepsilon)$ gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.
Comments67 pages, 2 figures
