Refined Belief Propagation Decoding of Sparse-Graph Quantum Codes

Abstract

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.

Degeneracy and Its Impact on the Decoding of Sparse Quantum Codes

“…All in all, in order for QLDPC codes to realize their full potential, means of minimizing the frequency with which end-to-end errors with identical syndromes and end-to-end errors with different syndromes occur are necessary. The former requires a degenerate decoder capable of considering the coset nature of stabilizer codes while the latter can be achieved by making use of improved SPA decoding strategies [16], [18], [53], [65], [113]- [117]. We summarize these strategies in what remains of this section.…”

“…Later on, in [18], a modified decoder for dual-containing CSS QLDPC codes which also led to performance improvements was introduced. Recently, in [114]- [117], a number of new and improved decoding strategies have been proposed. A timeline of the most significant contributions to the field of modifed QLDPC decoding is provided in table 1.…”

“…With the exceptions of the decoders proposed in [18], [20], [117], all of these heuristic methods operate according to the same working principle: prior to applying any changes to the decoder, the original qubit-wise BP decoder must fail. Generally, these modified decoding processes begin by running a standard unaltered version of the qubit-wise BP decoder.…”

“…Also in [114], the authors apply the two proposed approaches in unison, so that when one fails the other is applied and viceversa, a technique called the combined decoder. Finally, the most recent strategies of [115]- [117] focus more on postprocessing and direct modifications to the decoding algorithm.…”

Trapping Sets of Quantum LDPC Codes

“…The existing QLDPC code literature primarily focuses on constructing asymptotically good code families with improved minimum distance scaling with the block length and higher code rates, as well as on designing better iterative decoding algorithms [6][7][8][9][10][11][12]. However, QLDPC codes implemented in practical QEC systems will be of finite length and will exhibit performance degradation due to the failure of iterative decoders to converge to a correct error pattern.…”