Average mixing in quantum walks of reversible Markov chains

Pub Date : 2024-08-08 DOI:10.1016/j.disc.2024.114196
Julien Sorci
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Abstract

The Szegedy quantum walk is a discrete time quantum walk model which defines a quantum analogue of any Markov chain. The long-term behavior of the quantum walk can be encoded in a matrix called the average mixing matrix, whose columns give the limiting probability distribution of the walk given an initial state. We define a version of the average mixing matrix of the Szegedy quantum walk which allows us to more readily compare the limiting behavior to that of the chain it quantizes. We prove a formula for our mixing matrix in terms of the spectral decomposition of the Markov chain and show a relationship with the mixing matrix of a continuous quantum walk on the chain. In particular, we prove that average uniform mixing in the continuous walk implies average uniform mixing in the Szegedy walk. We conclude by giving examples of Markov chains of arbitrarily large size which admit average uniform mixing in both the continuous and Szegedy quantum walk.

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可逆马尔可夫链量子行走中的平均混合
塞格迪量子行走是一种离散时间量子行走模型,它定义了任何马尔可夫链的量子类似物。量子行走的长期行为可以用一个称为平均混合矩阵的矩阵来编码,该矩阵的列给出了给定初始状态下量子行走的极限概率分布。我们定义了一种塞格迪量子行走的平均混合矩阵,它能让我们更容易地将其极限行为与量子化链的极限行为进行比较。我们用马尔可夫链的谱分解证明了混合矩阵的公式,并展示了与链上连续量子行走的混合矩阵之间的关系。特别是,我们证明了连续行走的平均均匀混合意味着塞格迪行走的平均均匀混合。最后,我们举例说明了任意大的马尔科夫链,这些马尔科夫链在连续量子漫步和塞格迪量子漫步中都允许平均均匀混合。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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