Quantum XYZ Stabilizer Codes

Quantum error correction is dominated by the Calderbank-Shor-Steane (CSS) framework, which relies on mutually orthogonal X-type and Z-type stabilizer generators. While CSS codes are algebraically convenient, they restrict the available stabilizer configurations. This paper investigates whether relaxing this restriction by including Y-type stabilizer generators—forming what the authors call 'quantum XYZ stabilizer codes'—can improve the minimum distance and logical error rate of quantum low-density parity-check (qLDPC) codes in the finite-length regime.

The authors define XYZ stabilizer codes using three pairwise orthogonal binary parity-check matrices. A key challenge is that simply adding Y-type checks does not guarantee a genuinely non-CSS code, as the resulting stabilizer group might still be representable by a CSS generating set. The researchers provide algebraic and rank-based conditions to identify when these Y-type checks are redundant versus when they create a genuinely new, non-CSS structure. They further derive upper and lower bounds on the quantum minimum distance and test their constructions using quaternary belief propagation (BP4) decoding on depolarizing channels.

The study demonstrates that XYZ codes can effectively enlarge the design space for quantum codes. By analyzing the interaction between the three component codes, the authors show that Y-type constraints can increase the minimum distance compared to CSS codes derived from similar parameters. Numerical simulations confirm that these XYZ qLDPC instances can outperform representative CSS codes of similar block length and rate, providing empirical evidence that the additional design freedom translates into better error-correction performance at finite lengths.