Quantum Algorithms and Covering Spaces

It’s been recently demonstrated that quantum walks on graphs can solve certain computational problems faster than any classical algorithm. Therefore it is desirable to quantify those purely combinatorial properties of graphs which quantum walks take advantage of and try and separate them from those properties due to the encoding of the problem. In this paper we isolate the combinatorial property responsible (at least in part) for the computational speedups recently observed. We find that continuous-time quantum walks can exploit the covering space property of certain graphs. We formalise the notion of graph covering spaces. Then 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. We argue that if Y πN −−→ XN πN−1 −−−−→ XN−1 πN−2 −−−−→ · · · π1 −→ X1 is a tower of graph covering spaces satisfying certain uniformity and growth conditions then there exists an efficient quantum gate sequence simulating the walk. For the case of the walk on the m-torus graph T on 2 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 [email protected] [email protected]. Present address: Department of Mathematics and Department of Computer Science, University of York, York YO10 5DD, UK