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## On optimal quantum codes (2004)

Venue: | Int. J. Quantum Inform |

Citations: | 33 - 3 self |

### Citations

2645 |
The Theory of Error-Correcting Codes
- MacWilliams, Sloane
- 1977
(Show Context)
Citation Context ...ch are composed of subsystems of dimension pm, where p is prime and m ∈ N. As a shorthand, we will use the term “qudit”. In the theory of classical error-correcting codes it is well known that by increasing the size of the underlying alphabet, codes with better parameters can be constructed.1,2 We will show that the same is true for quantum error-correcting codes. Quantum codes for qudit systems have been studied before3,4,5,6 including efficient algorithms for encoding these codes.7 It is known that codes encoding one qudit into five qudits which are capable to correct one error, denoted by [[5, 1, 3]]q, exist for quantum systems of any dimension.8 In general, by [[n, k, d]]q we will denote a quantum error-correcting code (QECC) which encodes k qudits of a q-dimensional quantum system into n qudits. The parameter d is the minimum distance of the code. A QECC with minimum distance d can be used to detect errors that involve Accepted for publication in the International Journal of Quantum Information. This work was presented in part at the ERATO Conference on Quantum Information Science (Kyoto, Japan, 2003). 1 2 M. Grassl, Th. Beth, M. Rotteler at most d−1 of the n subsystems. Alternatively... |

1338 | The Magma algebra system. I. The user language - Bosma, Cannon, et al. - 1997 |

348 | Good quantum error-correcting codes exist, Phys - Calderbank, Shor - 1996 |

228 | Fault-tolerant quantum computation with constant error
- Aharonov, Ben-Or
- 1997
(Show Context)
Citation Context ...ch are composed of subsystems of dimension pm, where p is prime and m ∈ N. As a shorthand, we will use the term “qudit”. In the theory of classical error-correcting codes it is well known that by increasing the size of the underlying alphabet, codes with better parameters can be constructed.1,2 We will show that the same is true for quantum error-correcting codes. Quantum codes for qudit systems have been studied before3,4,5,6 including efficient algorithms for encoding these codes.7 It is known that codes encoding one qudit into five qudits which are capable to correct one error, denoted by [[5, 1, 3]]q, exist for quantum systems of any dimension.8 In general, by [[n, k, d]]q we will denote a quantum error-correcting code (QECC) which encodes k qudits of a q-dimensional quantum system into n qudits. The parameter d is the minimum distance of the code. A QECC with minimum distance d can be used to detect errors that involve Accepted for publication in the International Journal of Quantum Information. This work was presented in part at the ERATO Conference on Quantum Information Science (Kyoto, Japan, 2003). 1 2 M. Grassl, Th. Beth, M. Rotteler at most d−1 of the n subsystems. Alternatively... |

82 | Nonbinary quantum stabilizer codes,”
- Ashikhmin, Knill
- 2001
(Show Context)
Citation Context ...ch are composed of subsystems of dimension pm, where p is prime and m ∈ N. As a shorthand, we will use the term “qudit”. In the theory of classical error-correcting codes it is well known that by increasing the size of the underlying alphabet, codes with better parameters can be constructed.1,2 We will show that the same is true for quantum error-correcting codes. Quantum codes for qudit systems have been studied before3,4,5,6 including efficient algorithms for encoding these codes.7 It is known that codes encoding one qudit into five qudits which are capable to correct one error, denoted by [[5, 1, 3]]q, exist for quantum systems of any dimension.8 In general, by [[n, k, d]]q we will denote a quantum error-correcting code (QECC) which encodes k qudits of a q-dimensional quantum system into n qudits. The parameter d is the minimum distance of the code. A QECC with minimum distance d can be used to detect errors that involve Accepted for publication in the International Journal of Quantum Information. This work was presented in part at the ERATO Conference on Quantum Information Science (Kyoto, Japan, 2003). 1 2 M. Grassl, Th. Beth, M. Rotteler at most d−1 of the n subsystems. Alternatively... |

81 | Nonbinary quantum codes,” - Rains - 1999 |

76 | Error Correcting Codes in Quantum Theory,” Physical Review Letters, - Steane - 1996 |

55 | Fault-Tolerant Quantum Computation with Higher-Dimensional Systems,” - Gottesman - 1998 |

17 | Information Rates Achievable with Algebraic Codes on Quantum Discrete Memoryless Channels,”
- Hamada
- 2003
(Show Context)
Citation Context ... which encodes k qudits of a q-dimensional quantum system into n qudits. The parameter d is the minimum distance of the code. A QECC with minimum distance d can be used to detect errors that involve Accepted for publication in the International Journal of Quantum Information. This work was presented in part at the ERATO Conference on Quantum Information Science (Kyoto, Japan, 2003). 1 2 M. Grassl, Th. Beth, M. Rotteler at most d−1 of the n subsystems. Alternatively, one can correct errors that involve less than d/2 subsystems. Recently it was shown that optimal quantum codes with parameters [[6, 2, 3]]p and [[7, 3, 3]]p exist for all primes p ≥ 3 (see Ref. 9). There also exist quantum codes [[p, 1, (p + 1)/2]]p encoding one qudit into many qudits which are capable to correct more than one error.3 We show that many more optimal quantum codes exist. Note that in this paper we consider only codes of finite length, and not the asymptotic performance of codes when the length tends to infinity (for this, see, e. g., Ref. 6). First we recall basic constructions of QECCs from classical codes.5,10,11 Then we present families of optimal classical codes suitable for these constructions. In Section 4 ... |

15 |
Efficient Quantum Circuits for Non-Qubit Quantum Error-correcting Codes,”
- Grassl, Rotteler, et al.
- 2003
(Show Context)
Citation Context ...qudits of a q-dimensional quantum system into n qudits. The parameter d is the minimum distance of the code. A QECC with minimum distance d can be used to detect errors that involve Accepted for publication in the International Journal of Quantum Information. This work was presented in part at the ERATO Conference on Quantum Information Science (Kyoto, Japan, 2003). 1 2 M. Grassl, Th. Beth, M. Rotteler at most d−1 of the n subsystems. Alternatively, one can correct errors that involve less than d/2 subsystems. Recently it was shown that optimal quantum codes with parameters [[6, 2, 3]]p and [[7, 3, 3]]p exist for all primes p ≥ 3 (see Ref. 9). There also exist quantum codes [[p, 1, (p + 1)/2]]p encoding one qudit into many qudits which are capable to correct more than one error.3 We show that many more optimal quantum codes exist. Note that in this paper we consider only codes of finite length, and not the asymptotic performance of codes when the length tends to infinity (for this, see, e. g., Ref. 6). First we recall basic constructions of QECCs from classical codes.5,10,11 Then we present families of optimal classical codes suitable for these constructions. In Section 4 we address the pr... |

8 | Quantum codes [[6, 2, 3]]p and [[7, 3, 3]]p (p ≥ 3) exist - Feng - 2002 |

7 | Five quantum register error correction code for higher spin systems,” - Chau - 1997 |