量子XYZ稳定器码
Quantum XYZ Stabilizer Codes
AI总结：
研究引入量子XYZ稳定器码，其校验矩阵由三个两两正交的二元校验矩阵构建。刻画了其坍缩情况，推导量子最小距离界，包含已知非CSS拓扑码，产生稀疏qLDPC构造，模拟显示该码在有限长度下性能优于CSS qLDPC。
AI中文摘要：
稳定器码通常在Calderbank--Shor--Steane（CSS）框架内构建，由两个相互正交的二元经典码定义X和Z型稳定器生成元。虽然这种结构在代数上很方便，但额外的非CSS约束可能有助于抑制低权重逻辑算子并提高有限长度下的解码性能。因此引入量子XYZ稳定器码，其校验矩阵由与X、Y和Z型稳定器生成元相关的三个两两正交的二元校验矩阵构建。一个重要点是XYZ码实例不一定自动是非CSS的：相同的稳定器群可能允许一个CSS生成集。我们刻画了这种坍缩，得到了判定Y型校验何时冗余以及何时定义真正非CSS稳定器约束的代数和秩条件。我们还推导了量子最小距离的上下界，包括混合泡利逻辑算子的界。该新框架包含一个已知的非CSS拓扑码，即XYZ²六边形码，还从相交子集和准二元码族产生了稀疏的有限长度量子低密度校验（qLDPC）构造。在去极化码容量噪声和四元置信传播解码下的模拟表明，所提出的XYZ qLDPC实例在具有相似有限长度参数时可以优于代表性的CSS qLDPC实例。
英文摘要：
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.
