Quantum Coding Bounds and a Closed-Form Approximation of the Minimum Distance Versus Quantum Coding Rate

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 <inline-formula> <tex-math notation="LaTeX">$n = 128$ </tex-math></inline-formula>, the minimum distance is bounded by <inline-formula> <tex-math notation="LaTeX">$11 < d < 22$ </tex-math></inline-formula>, while our formulation yields a minimum distance of <inline-formula> <tex-math notation="LaTeX">$d = 16$ </tex-math></inline-formula> for the above-mentioned code. Ultimately, our contributions can be used for the characterization of quantum stabilizer codes.