  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 with O(exp(-\mu t))
for some \mu>0. The problem is to decide whether this difference is greater
than a exp(-\mu T) or smaller than b exp(-\mu T) after some time instant T,
where T is polylogarithmic and the difference a-b is inverse polylogarithmic in
the number of vertices. Since the probabilities differ only by an exponentially
small amount, an exponential number of trials would be necessary if one tried
to answer this question by running the walk itself.
  A modification of this problem, asking whether there exists a pair of nodes
for which the probability difference is at least a exp(-\mu T), is
QCMA-complete.
