A quantum walk is the quantum analogue of a random walk. While it is relatively well understood how quantum walks can speed up random walk hitting times, it is a long-standing open question to what extent quantum walks can speed up the spreading or mixing rate of random walks on graphs. In this expository paper, inspired by a blog post by Terence Tao, we describe a particular perspective on this question that derives quantum walks from the discrete wave equation on graphs. This yields a description of the quantum walk dynamics as simply applying a Chebyshev polynomial to the random walk transition matrix. This perspective decouples the problem from its quantum origin, and highlights connections to earlier (non-quantum) work and the use of Chebyshev polynomials in random walk theory as in the Varopoulos-Carne bound. We illustrate the approach by proving a weak limit of the quantum walk dynamics on the lattice. This gives a different proof of the quadratically improved spreading behavior of quantum walks on lattices.

翻译：量子游走是随机游走的量子对应物。尽管量子游走如何加速随机游走击中时间已相对清晰，但量子游走能在多大程度上加速图上随机游走的扩散速率或混合速率，仍是一个长期悬而未决的问题。在这篇综述性论文中，受陶哲轩博客文章的启发，我们从图上的离散波动方程出发，描述了对该问题的一种特定视角。这一视角将量子游走动力学简单地解释为对随机游走转移矩阵应用切比雪夫多项式。该视角将问题与其量子起源解耦，并凸显了与早期（非量子）工作以及随机游走理论中切比雪夫多项式应用（如Varopoulos-Carne界）的联系。我们通过在格点上证明量子游走动力学的弱极限来阐明该方法，从而为格点上量子游走扩散行为的二次加速提供了另一种证明。