标题： 通过非交换最优传输度量从吉布斯态中准最优采样 显示英文标题

标题： Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics

摘要： 我们研究在任意维度的超立方格点上从局部对易哈密顿量中采样和制备量子吉布斯态的问题。 我们证明，任何满足我们称之为矩阵值量子条件互信息（MCMI）衰减的聚类条件的此类吉布斯态可以在量子计算机上准最优地制备。 我们通过控制相应戴维斯演化在归一化量子一阶Wasserstein距离中的混合时间来实现这一点。 据我们所知，这是首次在量子动力学研究中使用这种非交换运输度量，并且是首次仅由显式聚类条件推导出准快速混合。 我们的结果基于此类系统的一个弱近似张量化和一个弱修改对数索博列夫不等式，以及一个新的通用弱运输成本不等式。 如果我们进一步假设热化动力学的局部间隙存在约束，那么我们可以得到超出两体相互作用范围的迹距离快速混合，从而扩展了仅覆盖最近邻情况的最新结果。 最后，我们表明具有有效局部哈密顿量的系统，如高温下的量子CSS码，满足MCMI衰减，因此可以高效地制备和采样。 显示英文摘要

摘要： We study the problem of sampling from and preparing quantum Gibbs states of local commuting Hamiltonians on hypercubic lattices of arbitrary dimension. We prove that any such Gibbs state which satisfies a clustering condition that we coin decay of matrix-valued quantum conditional mutual information (MCMI) can be quasi-optimally prepared on a quantum computer. We do this by controlling the mixing time of the corresponding Davies evolution in a normalized quantum Wasserstein distance of order one. To the best of our knowledge, this is the first time that such a non-commutative transport metric has been used in the study of quantum dynamics, and the first time quasi-rapid mixing is implied by solely an explicit clustering condition. Our result is based on a weak approximate tensorization and a weak modified logarithmic Sobolev inequality for such systems, as well as a new general weak transport cost inequality. If we furthermore assume a constraint on the local gap of the thermalizing dynamics, we obtain rapid mixing in trace distance for interactions beyond the range of two, thereby extending the state-of-the-art results that only cover the nearest neighbor case. We conclude by showing that systems that admit effective local Hamiltonians, like quantum CSS codes at high temperature, satisfy this MCMI decay and can thus be efficiently prepared and sampled from.