2006/10/27 by Dominik Janzing, Janzing, Dominik, Paweł Wocjan +2 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Complexity and Algorithms in Graphs #FOS: Physical sciences #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0610235
24 pages, 1 figure
openalex publication_date 2006/10/27 · arxiv created 2006/10/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(-μt)) for some μ>0. The problem is to decide whether this difference is greater than a exp(-μT) or smaller than b exp(-μT) 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(-μT), is QCMA-complete.