BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs

We describe two BQP-complete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the difference between the number of paths of length m from j to j and those which from i to j, where m is polylogarithmic in the number of vertices. The scale of the estimation accuracy is specified by some a priori known upper bound on the growth of these differences with increasing m. The problem remains BQP-hard for regular graphs with degree 4. The second problem is related to continuous-time classical random walks. The walk starts at some vertex j. The promise is that the difference of the probabilities of being at j and at i, respectively, decays e-mail: [email protected] e-mail: [email protected]