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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Vydané v:46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) s. 469 - 478
Hlavní autori: Bacon, D., Childs, A.M., van Dam, W.
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Jazyk:English
Vydavateľské údaje: IEEE 2005
Predmet:
Computer science
Fourier transforms
Lattices
Particle measurements
Performance evaluation
Physics
Polynomials
Quantum computing
Quantum mechanics
State estimation
ISBN:0769524680, 9780769524689
ISSN:0272-5428
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Abstract 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.
AbstractList 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.
Author Childs, A.M.
Bacon, D.
van Dam, W.
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Snippet 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...
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StartPage 469
SubjectTerms Computer science
Fourier transforms
Lattices
Particle measurements
Performance evaluation
Physics
Polynomials
Quantum computing
Quantum mechanics
State estimation
Title From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups
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