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31
Quantum mechanical algorithms for the nonabelian Hidden Subgroup Problem
, 2000
"... We give a short exposition of new and known results on the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer. ..."
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Cited by 73 (6 self)
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We give a short exposition of new and known results on the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.
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 51 (10 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.
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.
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
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 24 (2 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 and Quantum Computation Using Group Representations
 SIAM Journal on Computing
, 2003
"... The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonAbelian case is open; an efficient solution would give ..."
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Cited by 24 (3 self)
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The Hidden Subgroup Problem is the foundation of many quantum algorithms. An efficient solution is known for the problem over Abelian groups, employed by both Simon's algorithm and Shor's factoring and discrete log algorithms. The nonAbelian case is open; an efficient solution would give rise to an efficient quantum algorithm for Graph Isomorphism. We fully analyze a natural generalization of the Abelian case algorithm to the nonAbelian case. We show that the algorithm finds the normal core of the hidden subgroup, and that, in particular, normal subgroups can be found. We show, however, that this immediate generalization of the Abelian algorithm does not efficiently solve Graph Isomorphism. 1
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 21 (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 ReedSolomon Codes
 in Proceedings Applied Algebra, Algebraic Algorithms and ErrorCorrecting Codes
"... Abstract — We introduce a new class of quantum error–correcting codes derived from (classical) Reed– Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented. I. ..."
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Cited by 13 (4 self)
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Abstract — We introduce a new class of quantum error–correcting codes derived from (classical) Reed– Solomon codes over finite fields of characteristic two. Quantum circuits for encoding and decoding based on the discrete cyclic Fourier transform over finite fields are presented. I.
Discrete Cosine Transforms on Quantum Computers
 PROCEEDINGS OF THE 2ND INTERNATIONAL SYMPOSIUM ON IMAGE AND SIGNAL PROCESSING AND ANALYSIS
, 2001
"... A classical computer does not allow to calculate a discrete cosine transform on N points in less than linear time. This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and interference principles. In fact, we show that it ..."
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Cited by 10 (2 self)
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A classical computer does not allow to calculate a discrete cosine transform on N points in less than linear time. This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and interference principles. In fact, we show that it is possible to realize the discrete cosine transforms and the discrete sine transforms of size N N and types I,II,III, and IV with as little as O(log 2 N) operations on a quantum computer, whereas the known fast algorithms on a classical computer need O(N log N) operations.
Recent progress and applications in group FFTs
 NATO Advanced Study Institute on Computational Noncommutative Algebra and Applications
, 2003
"... The CooleyTukey FFT can be interpreted as an algorithm for the efficient computation of the Fourier transform for finite cyclic groups, a compact group (the circle), or the noncompact group of the real line. These are all commutative instances of a “Group FFT. ” We give a brief survey of some rece ..."
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Cited by 9 (0 self)
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The CooleyTukey FFT can be interpreted as an algorithm for the efficient computation of the Fourier transform for finite cyclic groups, a compact group (the circle), or the noncompact group of the real line. These are all commutative instances of a “Group FFT. ” We give a brief survey of some recent progress made in the direction of noncommutative generalizations and their applications.