arXiv 1908.035772019-08-09hep-thquant-ph

QFT中的复杂性度量和约束几何动作

Complexity measures in QFT and constrained geometric actions

Pablo Bueno, Javier M. Magan, C. S. Shahbazi

备注：: 57 pages, 1 figure

摘要：

我们研究的条件下，给定一个通用的量子系统，复杂性度量提供实际的下限电路的复杂性与一组量子门。我们的分析排除了非齐次成本函数-最近在文献中提出了许多例子。这种措施被证明是无关的电路复杂性一般，并产生严重违反劳氏约束在简单的情况下。在提供下限的指标中，我们的想法是选择那些产生最紧密的可能。这建立了成本函数的层次结构，并大大减少了候选复杂性度量的列表。特别是，该标准提出了一个规范的方式来处理处罚，包括在分配无限的成本不属于门集的方向。我们将讨论如何通过使用拉格朗日乘数来实现这一点。我们认为，一个幸存的成本函数定义了一个特别规范的概念，在这个意义上说：i）它直接从希尔伯特空间中的标准厄米特度量; ii）其相关的复杂性泛函是密切相关的基里洛夫的coadjoint轨道作用，提供了一个明确的实现“复杂性等于作用”的想法; iii）它产生于描述与每个希尔伯特空间正则相关的相空间中的量子动力学的“量子作用”的哈密顿-雅可比分析。最后，我们解释了这些结构如何提供一个自然的框架，在经典和量子系统中的混沌特征在平等的基础上，找到最小的测地线连接两个附近的轨迹，并描述如何复杂的措施是敏感的李雅普诺夫指数。

英文摘要：

We study the conditions under which, given a generic quantum system, complexity metrics provide actual lower bounds to the circuit complexity associated to a set of quantum gates. Inhomogeneous cost functions ---many examples of which have been recently proposed in the literature--- are ruled out by our analysis. Such measures are shown to be unrelated to circuit complexity in general and to produce severe violations of Lloyd's bound in simple situations. Among the metrics which do provide lower bounds, the idea is to select those which produce the tightest possible ones. This establishes a hierarchy of cost functions and considerably reduces the list of candidate complexity measures. In particular, the criterion suggests a canonical way of dealing with penalties, consisting in assigning infinite costs to directions not belonging to the gate set. We discuss how this can be implemented through the use of Lagrange multipliers. We argue that one of the surviving cost functions defines a particularly canonical notion in the sense that: i) it straightforwardly follows from the standard Hermitian metric in Hilbert space; ii) its associated complexity functional is closely related to Kirillov's coadjoint orbit action, providing an explicit realization of the ``complexity equals action'' idea; iii) it arises from a Hamilton-Jacobi analysis of the ``quantum action'' describing quantum dynamics in the phase space canonically associated to every Hilbert space. Finally, we explain how these structures provide a natural framework for characterizing chaos in classical and quantum systems on an equal footing, find the minimal geodesic connecting two nearby trajectories, and describe how complexity measures are sensitive to Lyapunov exponents.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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