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Heuristic construction of codeword stabilized codes

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Rigby, A, Olivier, JC ORCID: 0000-0002-7703-6357 and Jarvis, P ORCID: 0000-0002-5330-6789 2019 , 'Heuristic construction of codeword stabilized codes' , Physical Review A, vol. 100, no. 6 , pp. 1-17 , doi: 10.1103/PhysRevA.100.062303.

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Abstract

The family of codeword stabilized codes encompasses the stabilizer codes as well as many of the best known nonadditive codes. However, constructing optimal n-qubit codeword stabilized codes is made difficult by two main factors. The first of these is the exponential growth with n of the number of graphs on which a code can be based. The second is the NP-hardness of the maximum clique search required to construct a code from a given graph. We address the second of these issues through the use of a heuristic clique finding algorithm. This approach has allowed us to find ((9,97≤K≤100,2)) and ((11,387≤K≤416,2)) codes, which are larger than any previously known codes. To address the exponential growth of the search space, we demonstrate that graphs that give large codes typically yield clique graphs with a large number of nodes. The number of such nodes can be determined relatively efficiently, and we demonstrate that n-node graphs yielding large clique graphs can be found using a genetic algorithm. This algorithm uses a crossover operation based on spectral bisection that we demonstrate to be superior to more standard crossover operations. Using this genetic algorithm approach, we have found ((13, 18, 4)) and ((13, 20, 4)) codes that are larger than any previously known code. We also consider codes for the amplitude damping channel. We demonstrate that for n≤9, optimal codeword stabilized codes correcting a single amplitude damping error can be found by considering standard form codes that detect one of only three of the 3n possible equivalent error sets. By combining this error-set selection with the genetic algorithm approach, we have found ((11, 68)) and ((11, 80)) codes capable of correcting a single amplitude damping error and ((11, 4)), ((12, 4)), ((13, 8)), and ((14, 16)) codes capable of correcting two amplitude damping errors.

Item Type: Article
Authors/Creators:Rigby, A and Olivier, JC and Jarvis, P
Keywords: graph theory
Journal or Publication Title: Physical Review A
Publisher: American Physical Society
ISSN: 2469-9934
DOI / ID Number: 10.1103/PhysRevA.100.062303
Copyright Information:

Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0) https://creativecommons.org/licenses/by/4.0/

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