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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 1263 (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.
Good quantum error correcting codes exist
 REV. A
, 1996
"... A quantum errorcorrecting code is defined to be a unitary mapping (encoding) of k qubits (2state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used ..."
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Cited by 347 (9 self)
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A quantum errorcorrecting code is defined to be a unitary mapping (encoding) of k qubits (2state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum errorcorrecting codes are shown to exist with asymptotic rate k/n = 1 − 2H2(2t/n) where H2(p) is the binary entropy function −p log2 p − (1 − p)log2(1 − p). Upper bounds on this asymptotic rate are given.
Faulttolerant quantum computation
 In Proc. 37th FOCS
, 1996
"... It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information i ..."
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Cited by 264 (5 self)
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It has recently been realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties in realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, making long computations impossible. A further difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering long computations unreliable. However, these obstacles may not be as formidable as originally believed. For any quantum computation with t gates, we show how to build a polynomial size quantum circuit that tolerates O(1 / log c t) amounts of inaccuracy and decoherence per gate, for some constant c; the previous bound was O(1 /t). We do this by showing that operations can be performed on quantum data encoded by quantum errorcorrecting codes without decoding this data. 1.
Reliable quantum computers
 Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
, 1998
"... The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mist ..."
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Cited by 164 (3 self)
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The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 106 qubits, with a probability of error per quantum gate of order 106, would be a formidable factoring engine. Even a smaller lessaccurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence
An Introduction to Quantum Computing for NonPhysicists
 Los Alamos Physics Preprint Archive http://xxx.lanl.gov/abs/quantph/9809016
, 2000
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Efficient simulation of quantum systems by quantum computers
, 1998
"... We show that the time evolution of the wave function of a quantummechanical manyparticle system can be simulated precisely and efficiently on a quantum computer. The time needed for such a simulation is comparable to the time of a conventional simulation of the corresponding classical system, a per ..."
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Cited by 79 (0 self)
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We show that the time evolution of the wave function of a quantummechanical manyparticle system can be simulated precisely and efficiently on a quantum computer. The time needed for such a simulation is comparable to the time of a conventional simulation of the corresponding classical system, a performance which can’t be expected from any classical simulation of a quantum system. We then show how quantities of interest, like the energy spectrum of a system, can be obtained. We also indicate that ultimately the simulation of quantum field theory might be possible on large quantum computers.
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 65 (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...
QUANTUM STRATEGIES
, 1998
"... We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). We introduce these basic ideas in the context of a simple example, closely related to the traditional MATCHING PENNIES game. While not eve ..."
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Cited by 59 (0 self)
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We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). We introduce these basic ideas in the context of a simple example, closely related to the traditional MATCHING PENNIES game. While not every twoperson zerosum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player’s expected payoff. We show, however, that in our example a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms. We prove that in general a quantum strategy is always at least as good as a classical one, and furthermore that when both players use quantum strategies there need not be any equilibrium, but if both are allowed mixed quantum strategies there must be.
Toward An Architecture For Quantum Programming
, 2003
"... It is becoming increasingly clear that, if a useful device for quantum computation will ever be built, it will be embodied by a classical computing machine with control over a truly quantum subsystem, this apparatus performing a mixture of classical and quantum computation. This paper investigates ..."
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Cited by 58 (0 self)
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It is becoming increasingly clear that, if a useful device for quantum computation will ever be built, it will be embodied by a classical computing machine with control over a truly quantum subsystem, this apparatus performing a mixture of classical and quantum computation. This paper investigates a possible approach to the problem of programming such machines: a template high level quantum language is presented which complements a generic general purpose classical language with a set of quantum primitives.