dorsal/arxiv
View SchemaBQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs
| Authors | Dominik Janzing, Pawel Wocjan |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0610235 |
| URL | https://arxiv.org/abs/quant-ph/0610235 |
Abstract
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.
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"abstract": "We describe two BQP-complete problems concerning properties of sparse graphs\nhaving a certain symmetry. The graphs are specified by efficiently computable\nfunctions which output the adjacent vertices for each vertex. Let i and j be\ntwo given vertices. The first problem consists in estimating the difference\nbetween the number of paths of length m from j to j and those which from i to\nj, where m is polylogarithmic in the number of vertices. The scale of the\nestimation accuracy is specified by some a priori known upper bound on the\ngrowth of these differences with increasing m. The problem remains BQP-hard for\nregular graphs with degree 4.\n The second problem is related to continuous-time classical random walks. The\nwalk starts at some vertex j. The promise is that the difference of the\nprobabilities of being at j and at i, respectively, decays with O(exp(-\\mu t))\nfor some \\mu\u003e0. The problem is to decide whether this difference is greater\nthan a exp(-\\mu T) or smaller than b exp(-\\mu T) after some time instant T,\nwhere T is polylogarithmic and the difference a-b is inverse polylogarithmic in\nthe number of vertices. Since the probabilities differ only by an exponentially\nsmall amount, an exponential number of trials would be necessary if one tried\nto answer this question by running the walk itself.\n A modification of this problem, asking whether there exists a pair of nodes\nfor which the probability difference is at least a exp(-\\mu T), is\nQCMA-complete.",
"arxiv_id": "quant-ph/0610235",
"authors": [
"Dominik Janzing",
"Pawel Wocjan"
],
"categories": [
"quant-ph"
],
"title": "BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs",
"url": "https://arxiv.org/abs/quant-ph/0610235"
},
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