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14
Quantum algorithms for solvable groups
 In Proceedings of the 33rd ACM Symposium on Theory of Computing
, 2001
"... ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, r ..."
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Cited by 38 (1 self)
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ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomialtime quantum algorithms as well. Our algorithm works in the setting of blackbox groups, wherein none of these problems have polynomialtime classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.
Hidden translation and orbit coset in quantum computing
 IN PROC. 35TH ACM STOC
, 2003
"... We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently ..."
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Cited by 38 (6 self)
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We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z n p, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful selfreducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G. Our selfreducibility framework combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.
Limits on the Power of Quantum Statistical ZeroKnowledge
, 2003
"... In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK ..."
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Cited by 28 (4 self)
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In this paper we propose a definition for honest verifier quantum statistical zeroknowledge interactive proof systems and study the resulting complexity class, which we denote QSZK
THE HIDDEN SUBGROUP PROBLEM  REVIEW AND OPEN PROBLEMS
, 2004
"... An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on ..."
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Cited by 11 (1 self)
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An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it easier to absorb the background necessary to begin research on the Hidden Subgroup Problem. Proofs are provided which give very concrete algorithms and bounds for the finite abelian case with little outside references, and future directions are provided for the nonabelian case. This summary is current as of October 2004.
Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Cited by 6 (2 self)
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
An Efficient Quantum Algorithm for the Hidden Subgroup Problem over a Class of Semidirect Product Groups
, 2005
"... Abstract. In this paper, we consider the hidden subgroup problem (HSP) over the class of semidirect product groups Zn ⋊ Zq. The definition of the semidirect product depending on the choice of an homomorphism, we first analyze the different possibilities for this homomorphism in function of n and q ..."
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Cited by 3 (2 self)
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Abstract. In this paper, we consider the hidden subgroup problem (HSP) over the class of semidirect product groups Zn ⋊ Zq. The definition of the semidirect product depending on the choice of an homomorphism, we first analyze the different possibilities for this homomorphism in function of n and q. Then, we present a polynomialtime quantum algorithm solving the HSP over the groups of the form Zpr ⋊ Zp, where p is an odd prime, and finally extend it to the class of groups Zm pr ⋊ Zp. 1 Introduction and Main Results The Hidden Subgroup Problem (HSP) is the problem of finding a subgroup H hidden in a known group G using a function f: G → N, provided as an oracle, which is Hperiodic, i. e. constant on all the elements of G in the same coset of H in G, and with a different value on each coset. A quantum algorithm of running time polynomial in log G  is known when the group
An efficient quantum algorithm for some instances of the group isomorphism problem. Full version of the present paper. Available at http://arxiv.org/abs/1001.0608
"... Abstract. In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has co ..."
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Cited by 3 (1 self)
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Abstract. In this paper we consider the problem of testing whether two finite groups are isomorphic. Whereas the case where both groups are abelian is well understood and can be solved efficiently, very little is known about the complexity of isomorphism testing for nonabelian groups. Le Gall has constructed an efficient classical algorithm for a class of groups corresponding to one of the most natural ways of constructing nonabelian groups from abelian groups: the groups that are extensions of an abelian group A by a cyclic group Zm with the order of A coprime with m. More precisely, the running time of that algorithm is almost linear in the order of the input groups. In this paper we present a quantum algorithm solving the same problem in time polynomial in the logarithm of the order of the input groups. This algorithm works in the blackbox setting and is the first quantum algorithm solving instances of the nonabelian group isomorphism problem exponentially faster than the best known classical algorithms. 1.
Quantum Algorithms in group theory
, 2003
"... We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory. ..."
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Cited by 1 (0 self)
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We present a survey of quantum algorithms, primarily for an intended audience of pure mathematicians. We place an emphasis on algorithms involving group theory.