Spatial search and the Dirac equation

We consider the problem of searching a d-dimensional lattice of N sites for a
single marked location. We present a Hamiltonian that solves this problem in
time of order sqrt(N) for d>2 and of order sqrt(N) log(N) in the critical
dimension d=2. This improves upon the performance of our previous quantum walk
search algorithm (which has a critical dimension of d=4), and matches the
performance of a corresponding discrete-time quantum walk algorithm. The
improvement uses a lattice version of the Dirac Hamiltonian, and thus requires
the introduction of spin (or coin) degrees of freedom.