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A statistical and comparative study of quantum walks under weak measurements and weak values regimes

机译：在弱测量和弱值体制下的量子行走的统计和比较研究

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Quantum walks have been studied under several regimes. Motivated by experimental results on quantum weak measurements and weak values as well as by the need to develop new insights for quantum algorithm development, we are extending our knowledge by studying the behavior of quantum walks under the regime of quantum weak measurements and weak values of pre- and postselected measurements (QWWM hereinafter). In particular, we investigate the limiting position probability distribution and several statistical measures (such as standard deviation) of a QWWM on an infinite line, and compare such results with corresponding classical and quantum walks position probability distributions and statistical measures, stressing the differences provided by weak measurements and weak values with respect to results computed by using canonical observables. We start by producing a concise introduction to quantum weak values and quantum weak measurements. We then introduce definitions as well as both analytical and numerical results for a QWWM under Hadamard evolution and extend our analysis to quantum evolution ruled by general unitary operators. Moreover, we propose a definition and focus on the properties of mixing time of QWWM on an infinite line, followed by a comparison of known corresponding results for classical and quantum walks mixing times. We finish this paper by presenting a plausible experimental implementation of a QWWM.
机译：已经在几种情况下研究了量子行走。受关于量子弱测量和弱值的实验结果以及对开发量子算法开发的新见解的需求的激励，我们通过研究在量子弱测量和pre的弱值情况下的量子行走行为来扩展我们的知识。 -和后选的测量值（以下称为QWWM）。特别是，我们研究了无限行上QWWM的极限位置概率分布和几种统计量度（例如标准差），并将这些结果与相应的经典和量子行走位置概率分布和统计量度进行比较，强调了由相对于使用规范可观观测值计算得出的结果而言，测量结果较弱且值较弱。我们首先简要介绍量子弱值和量子弱测量。然后，我们介绍了Hadamard演化下的QWWM的定义以及分析和数值结果，并将我们的分析扩展到由一般unit算子控制的量子演化。此外，我们提出了一个定义并集中在无限线上QWWM的混合时间的性质，然后比较了经典和量子行走混合时间的已知对应结果。我们通过提出一个可行的QWWM实验实现来完成本文。

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