Quantum XYZ Stabilizer Codes

• URL: http://arxiv.org/abs/2607.14988v1
• Date: Thu, 16 Jul 2026 13:38:00 GMT
• Title: Quantum XYZ Stabilizer Codes
• Abstract summary: We introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise binary PCMs.<n>A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS.<n>We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators.
• Score: 5.383800628085301
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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