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Pittsburgh, Pennsylvania, USA

Oct. 23, 2005 to Oct. 25, 2005

ISBN: 0-7695-2468-0

pp: 469-478

Dave Bacon , Dave Bacon

Andrew M. Childs , Andrew M. Childs

Wim van Dam , Wim van Dam

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/SFCS.2005.38

ABSTRACT

<p>We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form Z^r _p xZp for fixed r (including the Heisenberg group, r = 2). In particular, our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.</p>

INDEX TERMS

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CITATION

Dave Bacon,
Andrew M. Childs,
Wim van Dam,
"From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups",

*FOCS*, 2005, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2005, pp. 469-478, doi:10.1109/SFCS.2005.38