Quantum XYZ Stabilizer Codes

2026-07-16

AI Summary

Traditional CSS-type quantum stabilizer codes struggle to effectively suppress low-weight logical operators at finite code lengths, limiting their error-correction performance. This work proposes the XYZ stabilizer code framework, which systematically constructs non-CSS quantum codes incorporating three orthogonal binary parity-check matrices corresponding to X, Y, and Z Pauli operators, thereby transcending the structural constraints of CSS codes. The authors introduce an algebraic rank condition to ensure the non-redundancy of Y-type stabilizers, unifying known non-CSS topological codes as special cases. Leveraging triorthogonal constructions, bounds on minimum distance, and a quaternary belief-propagation decoding algorithm, the resulting sparse XYZ qLDPC codes outperform comparable CSS codes under depolarizing noise. Furthermore, the study establishes upper and lower bounds on the minimum distance for hybrid Pauli logical operators.

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.

Problem

• quantum stabilizer codes
• non-CSS codes
• quantum LDPC codes
• minimum distance
• logical operators

Innovation

• XYZ stabilizer codes
• non-CSS quantum codes
• quantum LDPC codes
• minimum distance bounds
• Y-type stabilizers