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Trellis decoders are a general decoding technique first applied toqubit-based quantum error correction codes by Ollivier and Tillich in 2006.Here we improve the scalability and practicality of their theory, show that ithas strong structure, extend the results using classical coding theory as aguide, and demonstrate a canonical form from which the structural properties ofthe decoding graph may be computed. The resulting formalism is valid for anyprime-dimensional quantum system. The modified decoder works for any stabilizercode $S$ and separates into two parts: a one-time, offline computation whichbuilds a compact, graphical representation of the normalizer of the code,$S^\perp$, and a quick, parallel, online query of the resulting vertices usingthe Viterbi algorithm. We show the utility of trellis decoding by applying itto four high-density, length 20 stabilizer codes for depolarizing noise and thewell-studied Steane, rotated surface, and 4.8.8/6.6.6 color codes for $Z$-onlynoise. Numerical simulations demonstrate a 20\% improvement in thecode-capacity threshold for color codes with boundaries by avoiding the mappingfrom color codes to surface codes. We identify trellis edge number as a keymetric of difficulty of decoding, allowing us to quantify the advantage ofsingle-axis decoding for Calderbank-Steane-Shor codes and block-decoding forconcatenated codes.

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