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151
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.
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 349 (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.
Quantum Error Correction Via Codes Over GF(4)
, 1997
"... The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on s ..."
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Cited by 304 (18 self)
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The problem of finding quantumerrorcorrecting codes is transformed into the problem of finding additive codes over the field GF(4) which are selforthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
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.
Concentrating partial entanglement by local operations
, 1995
"... If two separated observers are supplied with entanglement, in the form of n pairs of particles in identical partlyentangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximall ..."
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Cited by 178 (6 self)
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If two separated observers are supplied with entanglement, in the form of n pairs of particles in identical partlyentangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximallyentangled pairs of particles, for example EinsteinPodolskyRosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves entropy of entanglement—the von Neumann entropy of the partial density matrix seen by either observer—with the yield of singlets approaching, for large n, the base2 entropy of entanglement of the initial partlyentangled pure state. Conversely, any pure or mixed entangled state of two systems can be produced by two classicallycommunicating separated observers, drawing on a supply of singlets as their sole source of entanglement. Recent results in quantum information theory have shed light on the channel resources needed for faithful transmission of quantum states, and the extent to which these resources
Quantum information theory
, 1998
"... We survey the field of quantum information theory. In particular, we discuss the fundamentals of the field, source coding, quantum errorcorrecting codes, capacities of quantum channels, measures of entanglement, and quantum cryptography. ..."
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Cited by 102 (2 self)
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We survey the field of quantum information theory. In particular, we discuss the fundamentals of the field, source coding, quantum errorcorrecting codes, capacities of quantum channels, measures of entanglement, and quantum cryptography.
Do we really understand quantum mechanics? Strange correlations, paradoxes, and theorems
 Am. J. Phys
, 2001
"... This article presents a general discussion of several aspects of our present understanding of quantum mechanics. The emphasis is put on the very special correlations that this theory makes possible: they are forbidden by very general arguments based on realism and local causality. In fact, these cor ..."
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Cited by 60 (1 self)
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This article presents a general discussion of several aspects of our present understanding of quantum mechanics. The emphasis is put on the very special correlations that this theory makes possible: they are forbidden by very general arguments based on realism and local causality. In fact, these correlations are completely impossible in any circumstance, except the very special situations designed by physicists especially to observe these purely quantum effects. Another general point that is emphasized is the necessity for the theory to predict the emergence of a single result in a single realization of an experiment. For this purpose, orthodox quantum mechanics introduces a special postulate: the reduction of the state vector, which comes in addition to the Schrödinger evolution postulate. Nevertheless, the presence in parallel of two evolution processes of the same object (the state vector) may be a potential source for conflicts; various attitudes that are possible
Authentication of Quantum Messages
, 2002
"... Authentication is a wellstudied area of classical cryptography: a sender A and a receiver B sharing a classical private key want to exchange a classical message with the guarantee that the message has not been modified or replaced by a dishonest party with control of the communication line. In th ..."
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Cited by 59 (9 self)
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Authentication is a wellstudied area of classical cryptography: a sender A and a receiver B sharing a classical private key want to exchange a classical message with the guarantee that the message has not been modified or replaced by a dishonest party with control of the communication line. In this paper we study the authentication of messages composed of quantum states.
Quantum Key Distribution and String Oblivious Transfer on Noisy Channels, Los Alamos preprint archive quantph/9606003
 Advances in Cryptology: Proceeding of Crypto ’96, Lecture Notes in Computer Science
"... Abstract. We prove the unconditional security of a quantum key distribution (QKD) protocol on a noisy channel against the most general attack allowed by quantum physics. We use the fact that in a previous paper we have reduced the proof of the unconditionally security of this QKD protocol to a proof ..."
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Cited by 52 (8 self)
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Abstract. We prove the unconditional security of a quantum key distribution (QKD) protocol on a noisy channel against the most general attack allowed by quantum physics. We use the fact that in a previous paper we have reduced the proof of the unconditionally security of this QKD protocol to a proof that a corresponding Quantum String Oblivious Transfer (StringQOT) protocol would be unconditionally secure against Bob if implemented on top of an unconditionally secure bit commitment scheme. We prove a lemma that extends a security proof given by Yao for a (one bit) QOT protocol to this StringQOT protocol. This result and the reduction mentioned above implies the unconditional security of our QKD protocol despite our previous proof that unconditionally secure bit commitment schemes are impossible. 1