Universal computation by quantum walk

In some of the earliest work on quantum mechanical computers, Feynman showed
how to implement universal quantum computation by the dynamics of a
time-independent Hamiltonian. I show that this remains possible even if the
Hamiltonian is restricted to be a sparse matrix with all entries equal to 0 or
1, i.e., the adjacency matrix of a low-degree graph. Thus quantum walk can be
regarded as a universal computational primitive, with any desired quantum
computation encoded entirely in some underlying graph. The main idea of the
construction is to implement quantum gates by scattering processes.