吉布斯重采样跃迁与结构化快速测量协议

中文摘要 AI 辅助

量子吉布斯采样算法推广了经典马尔可夫链蒙特卡洛（cMCMC）方法，并通过在混合时间$t_{\rm mix}$内实现局部耗散演化，在量子计算机上制备热态。由于量子测量会破坏已制备的状态，从系统中提取无偏信息的速率取决于底层热关联的恢复。相应的“吉布斯重采样”时间$t_{\rm res}$既不同于$t_{\rm mix}$，也不同于cMCMC采样相关的自相关时间。我们在有效经典模型中研究此问题，这些模型模拟对扩展子系统上局部可观测量进行破坏性量子测量的效应。当此类子系统被随机选择时，在具有非局域序参量的模型中，我们发现了“重采样跃迁”，其作为被测量位点比例$p$的函数，在系统线性尺寸$L$下，从较小$p$时的$t_{\rm res}\propto \log L$转变为较大$p$时的$t_{\rm res}\propto {\rm poly} L$。我们认为，在具有局域序参量的系统中，这种跃迁通常不存在，并将这一二分性与在Glauber动力学下破坏1-形式和0-形式对称性的初始状态的粗化之间的二分性联系起来。最后，我们设计了一种结构化快速测量协议，即使在非局域序参量存在的情况下也能避免缓慢的重采样，同时仅留下所有位点中可忽略不计的一小部分未被测量。

英文摘要

Quantum Gibbs sampling algorithms generalize classical Markov chain Monte Carlo (cMCMC) methods, and prepare thermal states on quantum computers by implementing local dissipative evolution for a mixing time $t_{\rm mix}$. Since quantum measurements disrupt the prepared state, the rate of extracting unbiased information from the system depends on the recovery of the underlying thermal correlations. The corresponding ''Gibbs resampling'' time $t_{\rm res}$ is distinct both from $t_{\rm mix}$ and from the autocorrelation time relevant to cMCMC sampling. We study this problem in effective classical models that emulate disruptive quantum measurements of local observables on extensive subsystems. When such subsystems are chosen randomly, in models with non-local order parameters we uncover a ''resampling transition'' as a function of the fraction of sites $p$ that are measured, from $t_{\rm res}\propto \log L$ for small $p$ to $t_{\rm res}\propto {\rm poly} L$ for large $p$, in systems of linear size $L$. We argue that this transition is generically absent in systems with local order parameters, and relate this dichotomy to one between the coarsening of initial states that break 1- and 0-form symmetries under Glauber dynamics. Finally, we devise a structured fast-measurement protocol that evades slow resampling even with non-local order parameters, while leaving only a vanishing fraction of all sites unmeasured.

发表机构

• Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory（鲁道夫·皮尔斯理论物理中心，克拉伦登实验室）

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