Quantum XYZ Stabilizer Codes
Provides a new non-CSS framework for constructing quantum LDPC codes with improved finite-length performance, relevant to quantum error correction.
The authors introduce quantum XYZ stabilizer codes, which use three pairwise orthogonal binary parity-check matrices for X, Y, and Z stabilizers, and show that finite-length instances can outperform comparable CSS quantum LDPC codes under depolarizing noise.
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.