Non-additive quantum codes from Goethals and Preparata codes

Grassl, Markus; Rötteler, Martin

doi:10.1109/itw.2008.4578694

Cited by 11 publications

(15 citation statements)

References 20 publications

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“…To express the [ [5,1,3]] stabilizer code [see Example 1] as a ((5, 2, 3)) CWS code in standard form, we explicitly construct alternative generators S i of the stabilizer S = G 1 , G 2 , G 3 , G 4 , Z [Eq. (24)] to contain only one X operator each. We obtain S 3 = G 1 G 2 Z = IZXZI and its four cyclic permutations.…”

Section: E Graph Statesmentioning

confidence: 99%

Structured error recovery for code-word-stabilized quantum codes

Li¹,

Dumer²,

Grassl³

et al. 2010

Phys. Rev. A

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Codeword stabilized (CWS) codes are, in general, non-additive quantum codes that can correct errors by an exhaustive search of different error patterns, similar to the way that we decode classical non-linear codes. For an n-qubit quantum code correcting errors on up to t qubits, this brute-force approach consecutively tests different errors of weight t or less, and employs a separate n-qubit measurement in each test. In this paper, we suggest an error grouping technique that allows to simultaneously test large groups of errors in a single measurement. This structured error recovery technique exponentially reduces the number of measurements by about 3 t times. While it still leaves exponentially many measurements for a generic CWS code, the technique is equivalent to syndrome-based recovery for the special case of additive CWS codes.

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Section: E Graph Statesmentioning

confidence: 99%

Structured error recovery for code-word-stabilized quantum codes

Li¹,

Dumer²,

Grassl³

et al. 2010

Phys. Rev. A

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“…In [15], [16], the framework of union stabilizer codes has been introduced. Starting with a stabilizer code C 0 = [[n, k, d 0 ]] q , a union stabilizer code is given by…”

Section: Background and Notationsmentioning

confidence: 99%

“…In [8] a modified generalized concatenation has been introduced which uses outer code A i of different lengths n i as well as different inner codes B (0) j . Example 6: Using the stabilizer code B (1) = [ [21,15,3]] 2 , we can decompose the full space B (0) = [[21, 21, 1]] 2 into 64 mutually orthogonal codes [ [21,15,3]] 2 . In order to construct a generalized concatenated quantum code of distance three, we need a classical distance-three code over an alphabet of size 64, e. g., the classical MDS code A 1 = [65, 63, 3] 2 6 , as well as the trivial code A 2 = [65, 65, 1] 2 21+15 .…”

Section: A Stabilizer Codesmentioning

confidence: 99%

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Generalized concatenation for quantum codes

Grassl¹,

Zeng²

2009

2009 IEEE International Symposium on Information Theory

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“…These codes encompass stabilizer codes as well as all good known nonadditive codes. In addition, the CWS framework provides a powerful method to construct good nonadditive QECCs in a systematical way, and many good codes outperforming the best possible stabilizer codes were found [16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Experimental implementation of a codeword-stabilized quantum code

et al. 2012

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A five-qubit codeword stabilized quantum code is implemented in a seven-qubit system using nuclear magnetic resonance (NMR). Our experiment implements a good nonadditive quantum code which encodes a larger Hilbert space than any stabilizer code with the same length and capable of correcting the same kind of errors. The experimentally measured quantum coherence is shown to be robust against artificially introduced errors, benchmarking the success in implementing the quantum error correction code. Given the typical decoherence time of the system, our experiment illustrates the ability of coherent control to implement complex quantum circuits for demonstrating interesting results in spin qubits for quantum computing.PACS numbers: 03.67. Pp, 03.67.Lx, 03.65.Wj INTRODUCTIONQuantum computers are promising to solve certain problems faster than classical computers [1]. The power of quantum computing relies on the coherence of quantum states. In implementation, however, the quantum devices are subject to errors, from inevitable cou-2 pling to the uncontrollable environment, or from other mechanisms such as imperfection in controlled operations. The errors damage the coherence, and consequently can reduce the computational ability of quantum computers. In order to protect quantum coherence, schemes of quantum error correction and fault-tolerant quantum computation have been developed [1][2][3][4][5][6][7]. Those schemes have greatly improved the long-term prospects for quantum computation technology.A quantum error correcting code (QECC) protects a K-dimensional Hilbert space (the code space) by encoding it into an n-qubit system, and is usually denoted by parameters ((n, K, d)), where d is called the distance of the code [8]. This n-qubit system is used in the process of quantum computing and hence subject to errors. A code of distance d is capable of correcting d − 1 erasure errors (i.e. loss of qubits at up to d − 1 known positions) or⌋ arbitrary errors (i.e. arbitrary errors on t qubits at unknown positions). At the end of the computation, the quantum code can be decoded to recover the quantum state in the original K-dimensional Hilbert space.In practice, one would always hope for a "good" QECC where more information is protected (larger K), while less physical resources are used (smaller n) plus more errors can be corrected (larger d). There are trade-offs among these three parameters, and one can readily develop upper bounds and lower bounds for the third parameter if two of them are fixed [7]. With increasing length n of the codes, in most of the cases there is a gap between these upper and lower bounds. Therefore, given two fixed parameters among the three, to find a code with the best possible value of the third parameter is one of the most important topics in studying the theory of QECC.Stabilizer codes, also known as additive codes, form an important class of QECCs, developed independently in [6,7] in the late 1990s. The construction of these codes is based on a simple method using Abelian groups, where the...

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Non-additive quantum codes from Goethals and Preparata codes

Abstract: Abstract-We extend the stabilizer formalism to a class of nonadditive quantum codes which are constructed from non-linear classical codes. As an example, we present infinite families of nonadditive codes which are derived from Goethals and Preparata codes.

Cited by 11 publications

References 20 publications

Structured error recovery for code-word-stabilized quantum codes

Structured error recovery for code-word-stabilized quantum codes

Generalized concatenation for quantum codes

Experimental implementation of a codeword-stabilized quantum code

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