arXiv:quant-ph/0604174 · Quantum Information and Computation

Quantum measurements for hidden subgroup problems with optimal sample complexity

Masahito Hayashi, A. Kawachi, H. Kobayashi

Published Mar 1, 2008• 35 citations

Abstract

One of the central issues in the hidden subgroup problem is to bound the sample complexity, i.e., the number of identical samples of coset states sufficient and necessary to solve the problem. In this paper, we present general bounds for the sample complexity of the identification and decision versions of the hidden subgroup problem. As a consequence of the bounds, we show that the sample complexity for both of the decision and identification versions is $\Theta(\log|\HH|/\log p)$ for a candidate set $\HH$ of hidden subgroups in the case \REVISE{where the candidate nontrivial subgroups} have the same prime order $p$, which implies that the decision version is at least as hard as the identification version in this case. In particular, it does so for the important \REVISE{cases} such as the dihedral and the symmetric hidden subgroup problems. Moreover, the upper bound of the identification is attained \REVISE{by a variant of the pretty good measurement}. \REVISE{This implies that the concept of the pretty good measurement is quite useful for identification of hidden subgroups over an arbitrary group with optimal sample complexity}.

DOI: 10.26421/qic8.3-4-8

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arXiv:quant-ph/0604174 · Quantum Information and Computation

Quantum measurements for hidden subgroup problems with optimal sample complexity

Masahito Hayashi, A. Kawachi, H. Kobayashi

Published Mar 1, 2008• 35 citations

arXiv HTML PDF

Abstract

\Theta(\log|\HH|/\log p)

for a candidate set

\HH

of hidden subgroups in the case \REVISE{where the candidate nontrivial subgroups} have the same prime order

p

, which implies that the decision version is at least as hard as the identification version in this case. In particular, it does so for the important \REVISE{cases} such as the dihedral and the symmetric hidden subgroup problems. Moreover, the upper bound of the identification is attained \REVISE{by a variant of the pretty good measurement}. \REVISE{This implies that the concept of the pretty good measurement is quite useful for identification of hidden subgroups over an arbitrary group with optimal sample complexity}.

DOI: 10.26421/qic8.3-4-8

Related papers

Quantum measurements for hidden subgroup problems with optimal sample complexity

HayashiMasahito, KawachiAkinori, KobayashiHirotada

Quantum Information and ComputationMar 1, 20081 citation

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This dissertation concerns the Hidden Subgroup Problem (HSP) in quantum computation. We explore certain facets of the problem which, we expect, will lead towards a general framework for understanding such problems. We first present an implementation of an optimal measurement, the Pretty Good Measurement (PGM), for the problem of the hidden subgroup problem over the dihedral group. This is a problem for which no efficient algorithm is known. This problem is relevant to the shortest vector in a lattice problem (SVP) in cryptography. We give a representation theoretic implementation of the PGM, achievable with only abelian transforms. The subset sum problem, an NP-complete problem, is needed in identification of certain subspaces. There are some novel ideas and techniques used in the implementation that make it realizable. This brings into focus how a measurement specified through a POVM (positive operator valued measure) is achievable through a basis change in a particular representation. This has connections to the HSP for other groups. We also study the question of optimal query selection to maximize the probability of subgroup identification. This part of the research complements a well-researched topic of finding optimal measurements in the spirit of state discrimination as applied to the hidden subgroup problem. We find the optimal query, the "character query", over a class of queries that are an equal superposition over the group, for oracles that take values in a finite abelian group. This query outperforms the query used in the "standard method" of the HSP. This approach has connections to other domains of quantum computation. We then extend the query selection idea to the case of multiple queries in the HSP and find examples in which this gives significant gains over the standard method. We consider the problems that the analysis of this more general problem entails, and outline a possible program to be followed. This is a part of continuing researchThis dissertation concerns the Hidden Subgroup Problem (HSP) in quantum computation. We explore certain facets of the problem which, we expect, will lead towards a general framework for understanding such problems. We first present an implementation of an optimal measurement, the Pretty Good Measurement (PGM), for the problem of the hidden subgroup problem over the dihedral group. This is a problem for which no efficient algorithm is known. This problem is relevant to the shortest vector in a lattice problem (SVP) in cryptography. We give a representation theoretic implementation of the PGM, achievable with only abelian transforms. The subset sum problem, an NP-complete problem, is needed in identification of certain subspaces. There are some novel ideas and techniques used in the implementation that make it realizable. This brings into focus how a measurement specified through a POVM (positive operator valued measure) is achievable through a basis change in a particular representation. This has connections to the HSP for other groups. We also study the question of optimal query selection to maximize the probability of subgroup identification. This part of the research complements a well-researched topic of finding optimal measurements in the spirit of state discrimination as applied to the hidden subgroup problem. We find the optimal query, the "character query", over a class of queries that are an equal superposition over the group, for oracles that take values in a finite abelian group. This query outperforms the query used in the "standard method" of the HSP. This approach has connections to other domains of quantum computation. We then extend the query selection idea to the case of multiple queries in the HSP and find examples in which this gives significant gains over the standard method. We consider the problems that the analysis of this more general problem entails, and outline a possible program to be followed. This is a part of continuing research

Jan 1, 20110 citations

\Theta(\log|\HH|/\log p)

for a candidate set

\HH

of hidden subgroups in the case \REVISE{where the candidate nontrivial subgroups} have the same prime order

p

O(\log(|G|)/ε)

copies of the state, but whether this scaling is optimal has remained open. We settle the complexity of the abelian StateHSP in both the previously studied sample model and a new query model, which is a stronger and operationally natural generalization that provides access to the state-preparation unitary and its inverse. In the query model, we give a time-efficient quantum algorithm using

O(\log(|G/H|)/\sqrtε)

forward and inverse queries, and prove a matching

Ω(\log(|G/H|)/\sqrtε)

lower bound which holds even in the stronger conjugate-query and controlled-query settings. By contrast, we show that in the sample model,

Θ(\log(|G/H|)/ε)

copies are both sufficient and information-theoretically necessary, even if one allows for arbitrary collective measurements. Thus, the quadratic improvement in

ε

genuinely arises from coherent access to the preparation circuit. As applications, we obtain faster algorithms for learning stabilizer groups, locating unentanglement, and identifying hidden translation symmetries.