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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.
Polynomialtime solution to the hidden subgroup problem for a class of nonabelian groups
, 1998
"... We present a family of nonabelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer. ..."
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Cited by 46 (0 self)
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We present a family of nonabelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.
The symmetric group defies strong Fourier sampling
, 2005
"... We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem. Specifically, we show that the hidden subgroup problem over the symmetric group cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state ..."
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Cited by 34 (10 self)
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We resolve the question of whether Fourier sampling can efficiently solve the hidden subgroup problem. Specifically, we show that the hidden subgroup problem over the symmetric group cannot be efficiently solved by strong Fourier sampling, even if one may perform an arbitrary POVM on the coset state. These results apply to the special case relevant to the Graph Isomorphism problem. 1 Introduction: the hidden subgroup problem Many problems of interest in quantum computing can be reduced to an instance of the Hidden Subgroup Problem (HSP). We are given a group G and a function f with the promise that, for some subgroup H ⊆ G, f is invariant precisely under translation by H: that is, f is constant on the cosets of H and takes distinct values on distinct cosets. We then wish to determine the subgroup H by querying f.
Generic quantum Fourier transforms
 In Proc. 15th ACMSIAM SODA
, 2004
"... The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by “quantizing” the separation of variables technique that has been so successful in the s ..."
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Cited by 33 (11 self)
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The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by “quantizing” the separation of variables technique that has been so successful in the study of classical Fourier transform computations. Specifically, this framework applies the existence of computable Bratteli diagrams, adapted factorizations, and Gel’fandTsetlin bases to offer efficient quantum circuits for the QFT over a wide variety a finite Abelian and nonAbelian groups, including all group families for which efficient QFTs are currently known and many new group families. Moreover, the method gives rise to the first subexponentialsize quantum circuits for the QFT over the linear groups GLk(q), SLk(q), and the finite groups of Lie type, for any fixed prime power q. 1
Optimal measurements for the dihedral hidden subgroup problem. arXiv:quantph/0501044
"... SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peerreviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for pap ..."
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Cited by 32 (4 self)
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SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peerreviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
Fast Quantum Fourier Transforms for a Class of nonabelian Groups
"... . An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvabl ..."
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Cited by 32 (0 self)
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. An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2 n rst (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of nonabelian 2groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2 n is O(n 2 ) in all cases. 1 Introduction Quantum algorithms are a recent subject and possibly of central importance in physics and computer science. It has been shown that there are problems on which a putative quantum computer could outper...
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 22 (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.
A Rosetta stone for quantum mechanics with an introduction to quantum computation
, 2002
"... Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading ..."
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Cited by 21 (8 self)
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Abstract. The purpose of these lecture notes is to provide readers, who have some mathematical background but little or no exposure to quantum mechanics and quantum computation, with enough material to begin reading
THE POWER OF STRONG FOURIER SAMPLING: QUANTUM ALGORITHMS FOR AFFINE GROUPS AND HIDDEN SHIFTS
, 2007
"... Many quantum algorithms, including Shor’s celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup H of a group G must be determined from a quantum state ψ over G that is uniformly supported on a left coset of H. These hidden s ..."
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Cited by 14 (1 self)
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Many quantum algorithms, including Shor’s celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which an unknown subgroup H of a group G must be determined from a quantum state ψ over G that is uniformly supported on a left coset of H. These hidden subgroup problems are typically solved by Fourier sampling: the quantum Fourier transform of ψ is computed and measured. When the underlying group is nonabelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong standard method, where full measurement (i.e., the row and column of the representation, in a suitably chosen basis, as well as its name) occurs. It has remained open whether the strong standard method is indeed stronger, that is, whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak, or any other known, method. In this article, we settle this question in the affirmative. We show that hidden subgroups H of the qhedral groups, i.e., semidirect products Zq ⋉Zp, where q  (p − 1), and in particular the affine groups Ap, can be informationtheoretically reconstructed using the strong standard method. Moreover, if H  = p/polylog(p), these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation. We