Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes

Trellis decoders are a general decoding technique first applied to qubit-based quantum error correction codes by Ollivier and Tillich in 2006. Here we improve the scalability and practicality of their theory, show that it has strong structure, extend the results using classical coding theory as a guide, and demonstrate a canonical form from which the structural properties of the decoding graph may be computed. The resulting formalism is valid for any prime-dimensional quantum system. The modified decoder works for any stabilizer code $S$ and separates into two parts: a one-time, offline computation which builds a compact, graphical representation of the normalizer of the code, $S^\perp$, and a quick, parallel, online query of the resulting vertices using the Viterbi algorithm. We show the utility of trellis decoding by applying it to four high-density, length 20 stabilizer codes for depolarizing noise and the well-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes for $Z$-only noise. Numerical simulations demonstrate a 20\% improvement in the code-capacity threshold for color codes with boundaries by avoiding the mapping from color codes to surface codes. We identify trellis edge number as a key metric of difficulty of decoding, allowing us to quantify the advantage of single-axis decoding for Calderbank-Steane-Shor codes and block-decoding for concatenated codes.
Comment: Improved writing in all but Section III. Proofs of technical lemmas sent to appendix. Threshold analysis for numerical results and new numerical examples added