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PolynomialTime Quantum Algorithms for Pell's Equation and the Principal Ideal Problem
 in Proceedings of the 34th ACM Symposium on Theory of Computing
, 2001
"... Besides Shor's polynomialtime quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more nonoracle pro ..."
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Cited by 108 (7 self)
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Besides Shor's polynomialtime quantum algorithms for factoring and discrete log, all progress in understanding when quantum algorithms have an exponential advantage over classical algorithms has been through oracle problems. Here we give efficient quantum algorithms for two more nonoracle problems. The first is Pell's equation. Given a positive nonsquare integer d, Pell's equation is x&sup2;  dy&sup2; = 1 and the goal is to find its integer solutions. Factoring integers reduces to finding integer solutions of Pell's equation, but a reduction in the other direction is not known and appears more difficult. The second problem is the principal ideal problem in real quadratic number fields. Solving this problem is at least as hard as solving Pell's equation, and is the basis of a cryptosystem which is broken by our algorithm. We also state some related open problems from the area of computational algebraic number theory.
QUANTUM ALGORITHMS FOR SOME HIDDEN SHIFT PROBLEMS
 SIAM J. COMPUT
, 2006
"... Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in th ..."
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Cited by 58 (3 self)
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Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
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 (12 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
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 32 (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.
Architectural implications of quantum computing technologies
 ACM Journal on Emerging Technologies in Computing Systems (JETC
, 2006
"... In this article we present a classification scheme for quantum computing technologies that is based on the characteristics most relevant to computer systems architecture. The engineering tradeoffs of execution speed, decoherence of the quantum states, and size of systems are described. Concurrency, ..."
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Cited by 27 (4 self)
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In this article we present a classification scheme for quantum computing technologies that is based on the characteristics most relevant to computer systems architecture. The engineering tradeoffs of execution speed, decoherence of the quantum states, and size of systems are described. Concurrency, storage capacity, and interconnection network topology influence algorithmic efficiency, while quantum error correction and necessary quantum state measurement are the ultimate drivers of logical clock speed. We discuss several proposed technologies. Finally, we use our taxonomy to explore architectural implications for common arithmetic circuits, examine the implementation of quantum error correction, and discuss clusterstate quantum computation.
Quantum property testing
 In Proceedings of 14th SODA
, 2003
"... Hein R"ohrig \Lambda \Lambda ..."
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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 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 22 (2 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