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304
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 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
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
Nonbinary quantum codes
 IEEE Trans. Inform. Theory
, 1999
"... Abstract. We present several results on quantum codes over general alphabets (that is, in which the fundamental units may have more than 2 states). In particular, we consider codes derived from finite symplectic geometry assumed to have additional global symmetries. From this standpoint, the analogu ..."
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Cited by 81 (1 self)
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Abstract. We present several results on quantum codes over general alphabets (that is, in which the fundamental units may have more than 2 states). In particular, we consider codes derived from finite symplectic geometry assumed to have additional global symmetries. From this standpoint, the analogues of CalderbankShorSteane codes and of GF(4)linear codes turn out to be special cases of the same construction. This allows us to construct families of quantum codes from certain codes over number fields; in particular, we get analogues of quadratic residue codes, including a singleerror correcting code encoding one letter in five, for any alphabet size. We also consider the problem of faulttolerant computation through such codes, generalizing ideas of Gottesman.
Nonbinary stabilizer codes over finite fields
 IEEE Trans. Inform. Theory
, 2006
"... One formidable difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. In past years, many good quantum errorcorrecting codes had been derived as binary stabilizer codes. Faulttolerant quantum comput ..."
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Cited by 50 (11 self)
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One formidable difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. In past years, many good quantum errorcorrecting codes had been derived as binary stabilizer codes. Faulttolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the wellknown notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper. 1
SparseGraph Codes for Quantum ErrorCorrection
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2004
"... We present sparsegraph codes appropriate for use in quantum errorcorrection. Quantum errorcorrecting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number ..."
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Cited by 49 (0 self)
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We present sparsegraph codes appropriate for use in quantum errorcorrection. Quantum errorcorrecting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparsegraph codes keep the number of quantum interactions associated with the quantum errorcorrection process small: a constant number per quantum bit, independent of the blocklength. Third, sparsegraph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum errorcorrecting codes.
A GroupTheoretic Framework for the Construction of Packings in Grassmannian Spaces
, 2002
"... By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this ..."
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Cited by 40 (11 self)
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By using totally isotropic subspaces in an orthogonal space Ω + (2i,2), several infinite families of packings of 2 kdimensional subspaces of real 2 idimensional space are constructed, some of which are shown to be optimal packings. A certain Clifford group underlies the construction and links this problem with BarnesWall lattices, Kerdock sets and quantumerrorcorrecting codes.