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

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

2006-10-27

arxiv.org/abs/quant-ph/0610235v2

Physics

Quantum Physics

24 pages, 1 figure

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-170146