We propose a new method called decoupling representation to represent Pauli operators as vectors over
GF(2), based on which we propose partially decoupled belief propagation and fully decoupled belief propagation decoding algorithm for quantum low density parity-check codes. These two algorithms have the capability to deal with the correlations between the
X part and the
Z part of the vectors in symplectic representation, which are introduced by Pauli
Y errors. Hence, they can not only apply to CSS codes, but also to non-CSS codes. Under the assumption that there is no measurement error, compared with traditional belief propagation algorithm in symplectic representation over
GF(2), within the same number of iterations, the decoding accuracy of partially decoupled belief propagation and fully decoupled belief propagation algorithm is significantly improved in pure
Y noise and depolarizing noise, which supports that decoding algorithms of quantum error correcting codes might have better performance in decoupling representation than in symplectic representation. The impressive performance of fully decoupled belief propagation algorithm might promote the realization of quantum error correcting codes in engineering.
Quantum stabilizer codes constructed from sparse matrices have good performance and can be efficiently decoded by belief propagation (BP). A conventional BP decoding algorithm treats binary stabilizer codes as additive codes over GF(4). This algorithm has a relatively complex process of handling check-node messages, which incurs higher decoding complexity. Moreover, BP decoding of a stabilizer code usually suffers a performance loss due to the many short cycles in the underlying Tanner graph. In this paper, we propose a refined BP decoding algorithm for quantum codes with complexity roughly the same as binary BP. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but the passed node-to-node messages are single-valued, unlike the quaternary BP, where multivalued node-to-node messages are required. Furthermore, the techniques of message strength normalization can naturally be applied to these single-valued messages to improve the performance. Another observation is that the message-update schedule affects the performance of BP decoding against short cycles. We show that running BP with message strength normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error floor in computer simulation.
Codes based on sparse matrices have good performance and can be efficiently decoded by belief-propagation (BP). Decoding binary stabilizer codes needs a quaternary BP for (additive) codes over GF(4), which has a higher check-node complexity compared to a binary BP for codes over GF(2). Moreover, BP decoding of stabilizer codes suffers a performance loss from the short cycles in the underlying Tanner graph. In this paper, we propose a refined BP algorithm for decoding quantum codes by passing scalar messages. For a given error syndrome, this algorithm decodes to the same output as the conventional quaternary BP but with a check-node complexity the same as binary BP. As every message is a scalar, the message normalization can be naturally applied to improve the performance. Another observation is that the message-update schedule affects the BP decoding performance against short cycles. We show that running BP with message normalization according to a serial schedule (or other schedules) may significantly improve the decoding performance and error-floor in computer simulation.