Abstract
The tradeoff between the quantum coding rate and the associated error correction capability is characterized by the quantum coding bounds. The unique solution for this tradeoff 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 tradeoff between the quantum coding rate and the minimum distance of quantum stabilizer codes. For example, for a half-rate quantum stabilizer code having a code word length of $n = 128$ , the minimum distance is bounded by $11 , 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.
Highlights
Moore’s Law has remained valid for five decades, but based on its prediction at the time of writing the classical integrated circuits are expected to enter the nano-scale domain, where the laws of quantum mechanics prevail [1], [2]
We provide a survey of the existing quantum coding bounds found in the literature, along with their relationship to the existing quantum stabilizer code constructions
We have conducted a survey of quantum coding bounds, which describe the trade-off between the quantum coding rate and the error correction capability for a wide range of quantum stabilizer codes (QSCs) constructions
Summary
Moore’s Law has remained valid for five decades, but based on its prediction at the time of writing the classical integrated circuits are expected to enter the nano-scale domain, where the laws of quantum mechanics prevail [1], [2]. The family of entanglement-assisted quantum stabilizer codes (EA-QSCs) is capable of operating at a higher quantum coding rate than the unassisted QSC constructions at a given error correction capability, provided that errorfree maximally-entangled qubits have already been preshared [48], [49]. Against this background, our contributions are summarized as follows:.
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