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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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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.
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 45 (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.
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 39 (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
Quantum algorithms for algebraic problems
, 2008
"... Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational pro ..."
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Cited by 23 (1 self)
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Quantum computers can execute algorithms that dramatically outperform classical computation. As the bestknown example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum
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 19 (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.
The Quantum Fourier Transform and Extensions of the Abelian Hidden Subgroup Problem
, 2002
"... The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are ..."
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Cited by 12 (0 self)
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The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor’s celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitelygenerated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewisecontinuous function on ℜ. Finally, we show that this problem of periodfinding over ℜ, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, periodfinding
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 10 (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 4 (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
Linear and sublinear time algorithms for the basis of abelian groups
 Electronic Colloquium on Computational Complexity
, 2007
"... It is well known that every finite Abelian group G can be represented as a direct product of cyclic groups: G = G1◦G2◦ · · ·◦Gt, where each Gi is a cyclic group of size p j for some prime p and integer j ≥ 1. If ai is the generator of the cyclic group of Gi, i = 1, 2, · · · , t, then the element ..."
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Cited by 3 (1 self)
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It is well known that every finite Abelian group G can be represented as a direct product of cyclic groups: G = G1◦G2◦ · · ·◦Gt, where each Gi is a cyclic group of size p j for some prime p and integer j ≥ 1. If ai is the generator of the cyclic group of Gi, i = 1, 2, · · · , t, then the elements a1, a2, · · · , at are called a basis of G. In this paper, we first obtain an O(n)time deterministic algorithm for computing the basis of an Abelian group with n elements. The existing algorithms need O(n2) time by Chen and O(n1.5) time by Buchmann and Schmidt. We then derive an O(( ∑k i=1 p ni/2 i n 2 i)(log n) log log n)time randomized algorithm to compute the basis of Abelian group G of size n with factorization n = pn11 · · · p nt t, which is also a part of the input. It implies an O(n1/2(log n)3 log log n)time randomized algorithm to compute the basis of an Abelian group G of size n. It also implies that if n is an integer in {1, 2, · · ·,m}−G(m, c), then the basis of an Abelian group of size n can be computed in O((log n) c 2 +3 log log n)time, where c is any positive constant and G(m, c) is a subset of the small fraction of integers in {1, 2, · · ·,m} with