  Quantum computers are predicted to outperform classical ones for solving
partial differential equations, perhaps exponentially. Here we consider a
prototypical PDE - the heat equation in a rectangular region - and compare in
detail the complexities of ten classical and quantum algorithms for solving it,
in the sense of approximately computing the amount of heat in a given region.
We find that, for spatial dimension $d \ge 2$, there is an at most quadratic
quantum speedup using an approach based on applying amplitude estimation to an
accelerated classical random walk. However, an alternative approach based on a
quantum algorithm for linear equations is never faster than the best classical
algorithms.
