Limitations of quantum coset states for graph isomorphism

Sean Hallgren, Cristopher Moore, Martin Rötteler, Alexander Russell, Pranab Sen

Research output: Contribution to journalArticle

14 Scopus citations

Abstract

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore et al. [2005] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω (n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n, F pm) and Gn where G is finite and satisfies a suitable property.

Original languageEnglish (US)
Article number34
JournalJournal of the ACM
Volume57
Issue number6
DOIs
StatePublished - Oct 1 2010

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence

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