Quantum coding bounds and a closed-form approximation of the minimum distance versus quantum coding rate

Abstract

The trade-off between the quantum coding rate and the associated error correction capability is characterized by the quantum coding bounds. The unique solution for this trade-off does not exist, but the corresponding lower and the upper bounds can be found in the literature. In this treatise, we survey the existing quantum coding bounds and provide new insights into the classical to quantum duality for the sake of deriving new quantum coding bounds. Moreover, we propose an appealingly simple and invertible analytical approximation, which describes the trade-off between the quantum coding rate and the minimum distance of quantum stabilizer codes. For example, for a half-rate quantum stabilizer code having a codeword length of n = 128, the minimum distance is bounded by 11 < d < 22, while our formulation yields a minimum distance of d = 16 for the above-mentioned code. Ultimately, our contributions can be used for the characterization of quantum stabilizer codes.

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Identifiers

ORCID for Daryus Chandra: orcid.org/0000-0003-2406-7229

ORCID for Zunaira Babar: orcid.org/0000-0002-7498-4474

ORCID for Hung Nguyen: orcid.org/0000-0001-6349-1044

ORCID for Dimitrios Alanis: orcid.org/0000-0002-6654-1702

ORCID for Soon Ng: orcid.org/0000-0002-0930-7194

ORCID for Lajos Hanzo: orcid.org/0000-0002-2636-5214

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