Quantum Algorithms for Some Hidden Shift Problems
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.
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Information & Authors
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Published In
SIAM Journal on Computing
Pages: 763 - 778
ISSN (online): 1095-7111
Copyright
Copyright © 2006 Society for Industrial and Applied Mathematics.
History
Submitted: 15 July 2003
Accepted: 28 February 2006
Published online: 24 October 2006
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