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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.
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
Group Representations, Error Bases and Quantum Codes
 Los Alamos National Laboratory Report
, 1996
"... This report continues the discussion of unitary error bases and quantum codes begun in [8]. Nice error bases are characterized in terms of the existence of certain characters in a group. A general construction for error bases which are nonabelian over the center is given. The method for obtaining c ..."
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Cited by 41 (3 self)
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This report continues the discussion of unitary error bases and quantum codes begun in [8]. Nice error bases are characterized in terms of the existence of certain characters in a group. A general construction for error bases which are nonabelian over the center is given. The method for obtaining codes due to Calderbank et al. [2] is generalized and expressed purely in representation theoretic terms. The significance of the inertia subgroup both for constructing codes and obtaining the set of transversally implementable operations is demonstrated. Note: This report is preliminary. Please contact the author if you wish to be notified of updates. 1 Overview This report discusses the construction of quantum codes based on nice error bases [8]. The main conclusion is that much of the relevant theory can be cast in terms of representations of finite groups. It is shown that nice error bases are equivalent to the existence of an irreducible character with nonzero values only on the cent...
Parallel quantum computation and quantum codes.” quantph/9808027
"... Abstract. We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum errorcorrecting codes, ..."
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Cited by 39 (2 self)
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Abstract. We propose a definition of QNC, the quantum analog of the efficient parallel class NC. We exhibit several useful gadgets and prove that various classes of circuits can be parallelized to logarithmic depth, including circuits for encoding and decoding standard quantum errorcorrecting codes, or more generally any circuit consisting of controllednot gates, controlled πshifts, and Hadamard gates. Finally, while we note the Quantum Fourier Transform can be parallelized to linear depth, we conjecture that an even simpler ‘staircase ’ circuit cannot be parallelized to less than linear depth, and might be used to prove that QNC < QP. 1
Efficient computations of encodings for quantum error correction
 PHYS REV. A
, 1997
"... We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the codewords can be constructed. For an nqubit code whose stabilizer has d generators, the resulting gate array consists of O(nd) operations, and converts kqubit data (where k = n − ..."
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Cited by 28 (4 self)
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We show how, given any set of generators of the stabilizer of a quantum code, an efficient gate array that computes the codewords can be constructed. For an nqubit code whose stabilizer has d generators, the resulting gate array consists of O(nd) operations, and converts kqubit data (where k = n − d) into nqubit codewords.
Concatenated Quantum Codes
, 1996
"... One of the main problems for the future of practical quantum computing is to stabilize the computation against unwanted interactions with the environment and imperfections in the applied operations. Existing proposals for quantum memories and quantum channels require gates with asymptotically zero e ..."
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Cited by 23 (9 self)
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One of the main problems for the future of practical quantum computing is to stabilize the computation against unwanted interactions with the environment and imperfections in the applied operations. Existing proposals for quantum memories and quantum channels require gates with asymptotically zero error to store or transmit an input quantum state for arbitrarily long times or distances with fixed error. In this report a method is given which has the property that to store or transmit a qubit with maximum error ffl requires gates with error at most cffl and storage or channel elements with error at most ffl, independent of how long we wish to store the state or how far we wish to transmit it. The method relies on using concatenated quantum codes with hierarchically implemented recovery operations. The overhead of the method is polynomial in the time of storage or the distance of the transmission. Rigorous and heuristic lower bounds for the constant c are given. 1 Introduction Practica...