### Abstract

In this paper, we address the problem of designing a quantum encoder that maximizes the minimum output purity of a given decohering channel, where the minimum is taken over all possible pure inputs. This problem is cast as a max-min optimization problem with a rank constraint on an appropriately defined matrix variable. The problem is computationally very hard because it is nonconvex with respect to both the objective function (output purity) and the rank constraint. Despite this difficulty, we provide a tractable computational algorithm that produces the exact optimal solution for codespace of dimension 2. Moreover, this algorithm is easily extended to cover the general class of codespaces, in which case the solution is suboptimal in the sense that the suboptimized output purity serves as a lower bound of the exact optimal purity. The algorithm consists of a sequence of semidefinite programmings and can be performed easily. Two typical quantum error channels are investigated to illustrate the effectiveness of our method.

Original language | English |
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Article number | 012327 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 76 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jul 26 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Physics and Astronomy(all)

### Cite this

**Computational approach to quantum encoder design for purity optimization.** / Yamamoto, Naoki; Fazel, Maryam.

Research output: Contribution to journal › Article

*Physical Review A - Atomic, Molecular, and Optical Physics*, vol. 76, no. 1, 012327. https://doi.org/10.1103/PhysRevA.76.012327

}

TY - JOUR

T1 - Computational approach to quantum encoder design for purity optimization

AU - Yamamoto, Naoki

AU - Fazel, Maryam

PY - 2007/7/26

Y1 - 2007/7/26

N2 - In this paper, we address the problem of designing a quantum encoder that maximizes the minimum output purity of a given decohering channel, where the minimum is taken over all possible pure inputs. This problem is cast as a max-min optimization problem with a rank constraint on an appropriately defined matrix variable. The problem is computationally very hard because it is nonconvex with respect to both the objective function (output purity) and the rank constraint. Despite this difficulty, we provide a tractable computational algorithm that produces the exact optimal solution for codespace of dimension 2. Moreover, this algorithm is easily extended to cover the general class of codespaces, in which case the solution is suboptimal in the sense that the suboptimized output purity serves as a lower bound of the exact optimal purity. The algorithm consists of a sequence of semidefinite programmings and can be performed easily. Two typical quantum error channels are investigated to illustrate the effectiveness of our method.

AB - In this paper, we address the problem of designing a quantum encoder that maximizes the minimum output purity of a given decohering channel, where the minimum is taken over all possible pure inputs. This problem is cast as a max-min optimization problem with a rank constraint on an appropriately defined matrix variable. The problem is computationally very hard because it is nonconvex with respect to both the objective function (output purity) and the rank constraint. Despite this difficulty, we provide a tractable computational algorithm that produces the exact optimal solution for codespace of dimension 2. Moreover, this algorithm is easily extended to cover the general class of codespaces, in which case the solution is suboptimal in the sense that the suboptimized output purity serves as a lower bound of the exact optimal purity. The algorithm consists of a sequence of semidefinite programmings and can be performed easily. Two typical quantum error channels are investigated to illustrate the effectiveness of our method.

UR - http://www.scopus.com/inward/record.url?scp=34547409473&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34547409473&partnerID=8YFLogxK

U2 - 10.1103/PhysRevA.76.012327

DO - 10.1103/PhysRevA.76.012327

M3 - Article

VL - 76

JO - Physical Review A

JF - Physical Review A

SN - 2469-9926

IS - 1

M1 - 012327

ER -