The hidden subgroup problem and quantum computation using group representations

Sean Hallgren, Alexander Russell, Amnon Ta-Shma

Research output: Contribution to journalArticle

48 Scopus citations

Abstract

The hidden subgroup problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonabelian case, however, remains open; an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism. We fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup: in particular, normal subgroups can be determined. We show, however, that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism.

Original languageEnglish (US)
Pages (from-to)916-934
Number of pages19
JournalSIAM Journal on Computing
Volume32
Issue number4
DOIs
StatePublished - Jun 2003

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Mathematics(all)

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