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 rootN for d > 2 and of order rootN 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.