---
author:
- 'Chris Godsil, Natalie Mullin, Aidan Roy'
bibliography:
- 'IUM.bib'
title: Uniform Mixing and Association Schemes
---

Introduction
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Quantum walks are a quantum analogue of a random walk on a graph, and have recently been the subject of much investigation. For recent survey from a mathematician’s viewpoint, see [@godsil12]. A quantum walk can behave quite differently from a classical walk: for example Childs et al. [@childs03] found a graph in which the time to propagate from one node to another was sped up exponentially compared to any classical walk.

We describe the physical motivation for continuous-time quantum walks. Let $X$ be a graph with Laplacian matrix $L$, and let $p(t) = (p_1(t),\ldots,p_n(t))$ be a probability distribution over the vertices $\{1,\ldots,n\}$ of $X$, for each time $t$. Then $p(t)$ is a continuous-time random walk on $X$ if it satisfies the differential equation $$\frac{\mathrm{d}p(t)}{\mathrm{d}t} = -L p(t).$$ This is a continuous version of a discrete-time random walk, in which the Laplacian matrix is the transition matrix for a Markov process on the vertices. The solution to the differential equation is $$p(t) = e^{-Lt}p(0).$$ Since $L$ has rows which sum to $0$, $e^{-Lt}$ has rows which sum to $1$ and 