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PolynomialTime Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
 SIAM J. on Computing
, 1997
"... A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. ..."
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Cited by 1278 (4 self)
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A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.
Quantum algorithms revisited
 Proceedings of the Royal Society of London A
, 1998
"... Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multiparticle interference. We use th ..."
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Cited by 188 (16 self)
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Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multiparticle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision. 1.
Quantum cryptanalysis of hidden linear functions
 in Proceedings of Crypto’95, Lecture Notes in Comput. Sci. 963
, 1995
"... Abstract. Recently there has been a great deal of interest in the power of \Quantum Computers " [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor&a ..."
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Cited by 74 (0 self)
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Abstract. Recently there has been a great deal of interest in the power of \Quantum Computers " [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor's to obtain a general theorem about quantum polynomial time. We show that any cryptosystem based on what we refer to as a `hidden linear form ' can be broken in quantum polynomial time. Our results imply that the discrete log problem is doable in quantum polynomial time over any group including Galois elds and elliptic curves. Finally, we introduce the notion of `junk bits ' which are helpful when performing classical computations that are not injective. 1
Quantum computation of Fourier transforms over symmetric groups
 STOC'97
, 1997
"... Quantum computation of Fourier transforms over symmetric groups Many algorithmic developments in quantum complexity theory, including Shor’s celebrated algorithms for factoring and discrete logs, have made use of Fourier transforms over abelian groups. That is, at some point in the computation, the ..."
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Cited by 73 (0 self)
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Quantum computation of Fourier transforms over symmetric groups Many algorithmic developments in quantum complexity theory, including Shor’s celebrated algorithms for factoring and discrete logs, have made use of Fourier transforms over abelian groups. That is, at some point in the computation, the macline is in a superposition of states corresponding to elements of a finite abelian group G, and in quantum polynomial time (i.e., polynomial in log IGI), the machine is transformed according to the Fourier transform to a superposition of states corresponding to the irreducible representations of G. We give a quantum polynomial time algorithm for the Fourier transform for the symmetric groups Sn, adapting results obtained by Clausen and Diaconis–Rockmore to the quantum setting.
A lambda calculus for quantum computation
 SIAM Journal of Computing
"... The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propos ..."
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Cited by 65 (1 self)
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The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of Linear Logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.
A procedural formalism for quantum computing
, 1998
"... Despite many common concepts with classical computer science, quantum computing is still widely considered as a special discipline within the broad field of theoretical physics. One reason for the slow adoption of QC by the computer science community is the confusing variety of formalisms (Dirac not ..."
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Cited by 39 (2 self)
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Despite many common concepts with classical computer science, quantum computing is still widely considered as a special discipline within the broad field of theoretical physics. One reason for the slow adoption of QC by the computer science community is the confusing variety of formalisms (Dirac notation, matrices, gates, operators, etc.), none of which has any similarity with classical programming languages, as well as the rather “physical” terminology in most of the available literature. QCL (Quantum Computation Language) tries to fill this gap: QCL is a hight level, architecture independent programming language for quantum computers, with a syntax derived from classical procedural languages like C or Pascal. This allows for the complete implementation and simulation of quantum algorithms (including classical components) in one consistent formalism. Chapter 1 is an introduction into the basic concepts of quantum programming,
Introduction to Quantum Algorithms
, 2001
"... Abstract. These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. ..."
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Abstract. These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms.
Circuit for Shor’s algorithm using 2n+3 qubits
 54
, 2002
"... We try to minimize the number of qubits needed to factor an integer of n bits using Shor’s algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n 3 lg(n)) elementary quantum gates in a depth of O(n 3) to implement the factorization algorithm. The circuit is computable ..."
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Cited by 22 (0 self)
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We try to minimize the number of qubits needed to factor an integer of n bits using Shor’s algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n 3 lg(n)) elementary quantum gates in a depth of O(n 3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. 1
Quantum Computation
, 1998
"... In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which see ..."
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Cited by 17 (0 self)
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In the last few years, theoretical study of quantum systems serving as computational devices has achieved tremendous progress. We now have strong theoretical evidence that quantum computers, if built, might be used as a dramatically powerful computational tool, capable of performing tasks which seem intractable for classical computers. This review is about to tell the story of theoretical quantum computation. I left out the developing topic of experimental realizations of the model, and neglected other closely related topics which are quantum information and quantum communication. As a result of narrowing the scope of this paper, I hope it has gained the benefit of being an almost self contained introduction to the exciting field of quantum computation.
Testing ShiftEquivalence Of Polynomials By Deterministic, Probabilistic And Quantum Machines
 Theoretical Computer Science
, 1997
"... 10.08> n ) = f . Introduction In the paper we deal with the problem of testing, whether two given polynomials f; g 2 F [X 1 ; : : : ; Xn ] are shiftequivalent, i.e. there exists a shift ff 1 ; : : : ; ff n such that f(X 1 +ff 1 ; : : : ; Xn+ffn ) = g. Earlier,the issue of considering polynomia ..."
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Cited by 16 (1 self)
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10.08> n ) = f . Introduction In the paper we deal with the problem of testing, whether two given polynomials f; g 2 F [X 1 ; : : : ; Xn ] are shiftequivalent, i.e. there exists a shift ff 1 ; : : : ; ff n such that f(X 1 +ff 1 ; : : : ; Xn+ffn ) = g. Earlier,the issue of considering polynomials up to Supported by NSF grant CCR9424358. Typeset by A M ST E X 1 2 the shifts appeared in the context of the interpolation of shiftedsparse polynomials (see [7, 11, 8]), namely, the polynomials which become sparse after a suitable shift. We present the algorithms for computing the group S f;f of the shifts (fi 1 ; : : : ; fi n