We classify the time complexities of three important decoding problems for quantum stabilizer codes. First, regardless of the channel model, quantum bounded distance decoding is shown to be NP-hard, like what Berlekamp, McEliece and Tilborg did for classical binary linear codes in 1978. Then over the depolarizing channel, the decoding problems for finding a most likely error and for minimizing the decoding error probability are also shown to be NP-hard. Our results indicate that finding a polynomial-time decoding algorithm for general stabilizer codes may be impossible, but this, on the other hand, strengthens the foundation of quantum code-based cryptography.
In quantum coding theory, stabilizer codes are probably the most important class of quantum codes. They are regarded as the quantum analogue of the classical linear codes and the properties of stabilizer codes have been carefully studied in the literature. In this paper, a new but simple construction of stabilizer codes is proposed based on syndrome assignment by classical parity-check matrices. This method reduces the construction of quantum stabilizer codes to the construction of classical parity-check matrices that satisfy a specific commutative condition. The quantum stabilizer codes from this construction have a larger set of correctable error operators than expected. Its (asymptotic) coding efficiency is comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed, which have a larger set of correctable error operators than that of the quantum Reed-Muller codes developed previously in the literature. Quantum stabilizer codes inspired by classical quadratic residue codes are also constructed and some of which are optimal in terms of their coding parameters.
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