Quantum XYZ Stabilizer Codes
Information Theory
Summary
The authors introduce a new type of quantum error-correcting code called XYZ stabilizer codes, which use three sets of checks related to X, Y, and Z quantum operations instead of just two as in traditional CSS codes. They study when these XYZ codes are truly different from CSS codes and provide mathematical conditions to tell if the Y-type checks add new information. The authors also give bounds on how well these codes protect quantum information and demonstrate examples where these codes improve decoding performance compared to similar CSS-based codes. Their work includes both well-known codes and new constructions aimed at practical quantum error correction.
Authors
Alessio Baldelli, Davide Orsucci, Francisco Lázaro, Massimo Battaglioni
Abstract
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.