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171
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.
Mixed state entanglement and quantum error correction
 Phys. Rev., A
, 1996
"... Entanglement purification protocols (EPP) and quantum errorcorrecting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a bipartite mixed state M; with a QECC, an arbitra ..."
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Cited by 185 (7 self)
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Entanglement purification protocols (EPP) and quantum errorcorrecting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a bipartite mixed state M; with a QECC, an arbitrary quantum state ξ〉 can be transmitted at some rate Q through a noisy channel χ without degradation. We prove that an EPP involving oneway classical communication and acting on mixed state ˆ M(χ) (obtained by sharing halves of EPR pairs through a channel χ) yields a QECC on χ with rate Q = D, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts D1(M) and D2(M) that can be locally distilled from it by EPPs using one and twoway classical communication respectively, and give an exact expression for E(M) when M is Belldiagonal. While EPPs require classical communication, quantum channel coding does not, and we prove Q is not increased by adding oneway classical communication. However, both D and Q can be increased by adding twoway communication. We show that certain noisy quantum channels, for example a 50 % depolarizing channel, can be used for reliable transmission of quantum states if twoway communication is available, but cannot be used if only oneway communication is available. We exhibit a family of codes based on universal hashing able to achieve an asymptotic Q (or D) of 1S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5bit singleerrorcorrecting quantum block code. We prove that iff a QECC results in perfect fidelity for the case of the noerror error syndrome the QECC can be recast into a form where the encoder is the matrix inverse of the decoder. 1 PACS numbers: 03.65.Bz, 42.50.Dv, 89.70.+c 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 165 (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
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.
A Theory of Quantum ErrorCorrecting Codes
, 1996
"... Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. ..."
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Cited by 101 (11 self)
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Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of errorcorrecting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of eerrorcorrecting codes applies to eerrorcorrecting quantum codes, provided that the interaction is dominated by an identity component.
Nonbinary quantum stabilizer codes
 IEEE Transactions on Information Theory
, 2001
"... We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between selforthogonal codes over F4 and binary quantum codes to one between selforthogonal codes over Fq2 and qary quantum codes for any prime pow ..."
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Cited by 82 (3 self)
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We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between selforthogonal codes over F4 and binary quantum codes to one between selforthogonal codes over Fq2 and qary quantum codes for any prime power q. Index Terms — quantum stabilizer codes, nonbinary quantum codes, selforthogonal codes. 1