  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.
