Quantum Algorithms and Covering Spaces

Tobias J. Osborne, Simone Severini

Published 2004-03-17, updated 2004-05-11Version 3

In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed in some quantum walk algorithms. We find that continuous-time quantum walks can exploit the covering space property of certain graphs. We demonstrate that a quantum walk on a graph Y which covers a smaller graph X can be equivalent to a quantum walk on the smaller graph X. This equivalence occurs only when the walk begins on certain initial states, fibre-constant states, which respect the graph covering space structure. We illustrate these observations with walks on Cayley graphs; we show that walks on fibre-constant initial states for Cayley graphs are equivalent to walks on the induced Schreier graph. We also consider the problem of constructing efficient gate sequences simulating the time evolution of a continuous-time quantum walk. For the case of the walk on the m-torus graph T^m on 2^n vertices we construct a gate sequence which uses O(\poly(n)) gates which is independent of the time t the walk is simulated for (and so the sequence can simulate the walk for exponential times). We argue that there exists a wide class of nontrivial operators based on quantum walks on graphs which can be measured efficiently. We introduce a new general class of computational problems, HiddenCover, which includes a variant of the general hidden subgroup problem as a subclass. We argue that quantum computers ought to be able to utilise covering space structures to efficiently solve HiddenCover problems.