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A subexponential-time quantum algorithm for the dihedral hidden subgroup problem (2003)
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BibTeX
@MISC{Kuperberg03asubexponential-time,
author = {Greg Kuperberg},
title = {A subexponential-time quantum algorithm for the dihedral hidden subgroup problem},
year = {2003}
}
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Abstract
Abstract. We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity 2O(√log N). In this problem an oracle computes a function f on the dihedral group DN which is invariant under a hidden reflection in DN. By contrast, the classical query complexity of DHSP is O ( √ N). The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character transform, which in the case of DN is essentially the abelian quantum Fourier transform. This yields the name of a group representation of DN, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of DN. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.
Keyphrases
dihedral hidden subgroup problem subexponential-time quantum algorithm hidden shift problem favorable representation query complexity new qubit generated abelian group abelian quantum fourier transform hidden reflection unfavorable qubits quantum character transform group representation certain target representation hidden subgroup indecipherable qubit algorithm proceeds quantum algorithm direct measurement classical query complexity dihedral group dn
