Quantum walks, the discrete wave equation and Chebyshev polynomials
Abstract
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.
Cite
@article{arxiv.2402.07809,
title = {Quantum walks, the discrete wave equation and Chebyshev polynomials},
author = {Simon Apers and Laurent Miclo},
journal= {arXiv preprint arXiv:2402.07809},
year = {2024}
}