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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 1277 (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.
Algorithms for Quantum Computation: Discrete Logarithms and Factoring
, 1994
"... A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factol: It is not clear whether this is still true when quantum mechanics is taken into consider ..."
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Cited by 1111 (5 self)
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A computer is generally considered to be a universal computational device; i.e., it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factol: It is not clear whether this is still true when quantum mechanics is taken into consideration. Several researchers, starting with David Deutsch, have developed models for quantum mechanical computers and have investigated their computational properties. This paper gives Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored. These two problems are generally considered hard on a classical computer and have been used as the basis of several proposed cryptosystems. (We thus give the first examples of quantum cryptanulysis.)
Elementary Gates for Quantum Computation
, 1995
"... We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x, y)to(x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We in ..."
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Cited by 280 (11 self)
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We show that a set of gates that consists of all onebit quantum gates (U(2)) and the twobit exclusiveor gate (that maps Boolean values (x, y)to(x, x⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized DeutschToffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two and threebit quantum gates, the asymptotic number required for nbit DeutschToffoli gates, and make some observations about the number required for arbitrary nbit unitary operations.
Quantum measurements and the Abelian stabilizer problem
"... We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure ..."
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Cited by 196 (0 self)
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We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor’s results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.
TwoBit Gates Are Universal for Quantum Computation
, 1995
"... A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of threebit gates, by analogy to the universality of ..."
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Cited by 182 (9 self)
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A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of threebit gates, by analogy to the universality of the Toffoli threebit gate of classical reversible computing. Twobit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins. A "gearbox quantum computer" proposed here, based on the principles of atomic force microscopy, would permit the operation of such twobit gates in a physical system with very long phase breaking (i.e., quantum phase coherence) times. Simpler versions of the gearbox computer could be used to do experiments on EinsteinPodolskyRosen states and related entangled quantum states.
Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance
, 2001
"... The number of steps any classical computer requires in order to find the prime factors of an ldigit integer N increases exponentially with l, at least using algorithms [1] known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying th ..."
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Cited by 150 (4 self)
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The number of steps any classical computer requires in order to find the prime factors of an ldigit integer N increases exponentially with l, at least using algorithms [1] known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes [1, 2]. Quantum computers [3], however, could factor integers in only polynomial time, using Shor’s quantum factoring algorithm [4, 5, 6]. Although important for the study of quantum computers [7], experimental demonstration of this algorithm has proved elusive [8, 9, 10]. Here we report an implementation of the simplest instance of Shor’s algorithm: factorization of N=15 (whose prime factors are 3 and 5). We use seven spin1/2 nuclei in a molecule as quantum bits [11, 12], which can be manipulated with room temperature liquid state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to many quantum bit systems [13], but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum
Oracle quantum computing
 Brassard & U.Vazirani, Strengths and weaknesses of quantum computing
, 1994
"... \Because nature isn't classical, dammit..." ..."
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Cited by 115 (8 self)
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\Because nature isn't classical, dammit..."
An exact quantum polynomialtime algorithm for Simon's problem
 IN PROCEEDINGS OF THE 5TH ISRAELI SYMPOSIUM ON THEORY OF COMPUTING AND SYSTEMS (ISTCS'97
, 1997
"... We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upperbounded by a polynomial in the worst case. We show that a natural generalization of Simon’s problem can be solved in this way, whereas previous algorit ..."
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Cited by 96 (10 self)
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We investigate the power of quantum computers when they are required to return an answer that is guaranteed to be correct after a time that is upperbounded by a polynomial in the worst case. We show that a natural generalization of Simon’s problem can be solved in this way, whereas previous algorithms required quantum polynomial time in the expected sense only, without upper bounds on the worstcase running time. This is achieved by generalizing both Simon’s and Grover’s algorithms and combining them in a novel way. It follows that there is a decision problem that can be solved in exact quantum polynomial time, which would require expected exponential time on any classical boundederror probabilistic computer if the data is supplied as a black box.
Fast parallel circuits for the quantum Fourier transform
 PROCEEDINGS 41ST ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS’00)
, 2000
"... We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our ..."
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Cited by 72 (1 self)
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We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomialsize, in combination with classical polynomialtime pre and postprocessing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with boundederror probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the
Conventions for Quantum Pseudocode
, 1996
"... A few conventions for thinking about and writing quantum pseudocode are proposed. The conventions can be used for presenting any quantum algorithm down to the lowest level and are consistent with a quantum random access machine (QRAM) model for quantum computing. In principle a formal version of qua ..."
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Cited by 66 (1 self)
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A few conventions for thinking about and writing quantum pseudocode are proposed. The conventions can be used for presenting any quantum algorithm down to the lowest level and are consistent with a quantum random access machine (QRAM) model for quantum computing. In principle a formal version of quantum pseudocode could be used in a future extension of a conventional language. Note: This report is preliminary. Please let me know of any suggestions, omissions or errors so that I can correct them before distributing this work more widely. 1 Introduction It is increasingly clear that practical quantum computing will take place on a classical machine with access to quantum registers. The classical machine performs offline classical computations and controls the evolution of the quantum registers by initializing them in certain preparable states, operating on them with elementary unitary operations and measuring them when needed. Although architectures for an integrated machine are far f...