Exceptional Configurations for Search by Szegedy's Quantum Walk
Thomas G. Wong, Raqueline A. M. Santos · arXiv (Cornell University) · 2016
In seminal work, Szegedy gave a method for quantizing a classical Markov chain that has been especially useful for searching for marked vertices in graphs. We prove, however, that this algorithm fails when searching on the two-dimensional periodic square lattice with a marked diagonal, in the sense that the state of the system only evolves by acquiring minus signs. Then the state is a uniform probability distribution over the vertices for all time, which is equivalent to classically guessing and checking. Furthermore, we prove that this failure also occurs for any configuration of marked vertices on the one-dimensional cycle and for any configuration that reduces to it. Thus, Szegedy's quantum walk has a family of exceptional configurations for which it does not spread. Despite this, we exploit this stationary behavior to construct a quantum walk with an arbitrary polynomial speedup over the classical random walk, and it is the first example of a greater-than-quadratic speedup.