From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probabili...

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Bibliographic Details
Published in:46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) pp. 469 - 478
Main Authors: Bacon, D., Childs, A.M., van Dam, W.
Format: Conference Proceeding
Language:English
Published: IEEE 2005
Subjects:
Computer science
Fourier transforms
Lattices
Particle measurements
Performance evaluation
Physics
Polynomials
Quantum computing
Quantum mechanics
State estimation
ISBN:0769524680, 9780769524689
ISSN:0272-5428
Online Access:Get full text
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Summary:We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form /spl Zopf//sub p/ /sup r/ /spl times/ /spl Zopf//sub p/ fixed r (including the Heisenberg group, r = 2). In particular our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.
ISBN:0769524680
9780769524689
ISSN:0272-5428
DOI:10.1109/SFCS.2005.38