Quantum algorithms for some hidden shift problems

Wim Van Dam, Sean Hallgren, Lawrence Ip

Research output: Contribution to journalArticle

46 Scopus citations

Abstract

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

Original languageEnglish (US)
Pages (from-to)763-778
Number of pages16
JournalSIAM Journal on Computing
Volume36
Issue number3
DOIs
StatePublished - Dec 1 2006

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Mathematics(all)

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