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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
Resilient quantum computation
 Science
, 1998
"... This article is a short introduction to and review of the clusterstate model of quantum computation, in which coherent quantum information processing is accomplished via a sequence of singlequbit measurements applied to a fixed quantum state known as a cluster state. We also discuss a few novel pr ..."
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Cited by 96 (4 self)
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This article is a short introduction to and review of the clusterstate model of quantum computation, in which coherent quantum information processing is accomplished via a sequence of singlequbit measurements applied to a fixed quantum state known as a cluster state. We also discuss a few novel properties of the model, including a proof that the cluster state cannot occur as the exact ground state of any naturally occurring physical system, and a proof that measurements on any quantum state which is linearly prepared in one dimension can be efficiently simulated on a classical computer, and thus are not candidates for use as a substrate for quantum computation. Key words: quantum computation, cluster states, oneway quantum computer 1.
Quantum entanglement
, 2007
"... Contents All our former experience with application of quantum theory seems to say: what is predicted by quantum formalism must occur in laboratory. But the essence of quantum formalism — entanglement, recognized by Einstein, Podolsky, Rosen and Schrödinger — waited over 70 years to enter to laborat ..."
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Cited by 88 (1 self)
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Contents All our former experience with application of quantum theory seems to say: what is predicted by quantum formalism must occur in laboratory. But the essence of quantum formalism — entanglement, recognized by Einstein, Podolsky, Rosen and Schrödinger — waited over 70 years to enter to laboratories as a new resource as real as energy.
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
On the role of entanglement in quantum computational speedup
"... For any quantum algorithm operating on pure states we prove that the presence of multipartite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speedup over classical computation. Furthermore we prove ..."
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Cited by 79 (0 self)
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For any quantum algorithm operating on pure states we prove that the presence of multipartite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speedup over classical computation. Furthermore we prove that the algorithm can be classically efficiently simulated to within a prescribed tolerance η even if a suitably small amount of global entanglement (depending on η) is present. We explicitly identify the occurrence of increasing multipartite entanglement in Shor’s algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speedup might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum computational power. 1
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