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.

• Received 1 June 2004

DOI:https://doi.org/10.1103/PhysRevA.70.042312