Spatial search and the Dirac equation

Quantum mechanical computers can solve certain problems asymptotically faster than classical computers. One of the major advantages of quantum computation comes from a fast algorithm for the problem of finding a marked item among N items. Whereas a classical computer requires Θ(N) steps to solve this problem, Grover showed that a quantum computer can solve it using only O( √ N) steps [1], which is optimal [2]. To apply Grover’s algorithm, it must be possible to quickly perform a reflection about a superposition of all possible items. However, this may not be feasible if the items are distributed in space and the algorithm is restricted to access them by local moves. For example, if the items are arranged on a one-dimensional line, simply traveling from one end of the line to the other requires N moves, and a straightforward argument shows that no local algorithm, classical or quantum, can find a marked item in less time than Ω(N). But for other geometries, such as higher dimensional lattices, a quantum algorithm can conceivably achieve a speedup over the classical complexity of Θ(N). Recently, there has been considerable progress in understanding the spatial search problem for quantum computers. Aaronson and Ambainis gave an algorithm that finds a marked item in the optimal time O( √ N) for a lattice in d > 2 dimensions, and in time O( √ N logN) for a two-dimensional lattice [3]. Their algorithm is based on a carefully optimized recursive search of subcubes, which raises the question of whether a simpler algorithm could solve the problem just as quickly (or perhaps even faster in two dimensions). In particular, it is interesting to consider quantum walk algorithms, which only use local, time-independent dynamics. Two distinct kinds of quantum walk algorithms have been considered. In the continuous-time quantum walk [4], the algorithm is described by a time-independent Hamiltonian connecting adjacent sites. In the discrete-time quantum walk [5, 6, 7], the algorithm consists of repeated application