码字稳定量子码中魔态的仿射-稳定子阈值

中文摘要 AI 辅助

码字稳定量子码（CWS）为稳定子量子码和非加性量子纠错码提供了统一的图态描述。尽管每个独立的CWS码字态都是稳定子，但不同码字态的相干叠加可以是非稳定子的。我们开发了一种适配CWS的魔态见证框架，该框架可分离这种码字相干性，并将其转化为魔态鲁棒性的认证下界。主要结果是将自然CWS相干见证的稳定子阈值精确约化为经典CWS码字集合上的有限几何问题。对于一般加权叠加，该阈值通过枚举仿射交集和仿射二次相位计算；对于等权重叠加，相位优化消失，阈值完全由每个维度的仿射平面中可容纳的CWS码字数量决定。因此，在稳定子态上的量子优化问题转化为经典仿射关联问题。这种约化产生了一个固定参数算法、一个针对无限并稳定子族的解析下界，以及几个标准非加性CWS示例的精确有理证书。该框架还阐明了为什么精确枚举对于大型结构化族会失败，并将剩余任务确定为仿射交集问题。该结果提供了一种几何机制，通过该机制，非线性CWS码字集合可产生可认证的魔态。

英文摘要

Codeword-stabilized quantum codes give a unified graph-state description of stabilizer and nonadditive quantum error-correcting codes. Although each individual CWS word state is stabilizer, coherent superpositions of different word states can be nonstabilizer. We develop a CWS-adapted magic-witness framework that isolates this codeword coherence and converts it into certified lower bounds on robustness of magic. The main result is an exact reduction of the stabilizer threshold of a natural CWS coherence witness to a finite-geometric problem over the classical CWS word set. For general weighted superpositions, the threshold is computed by enumerating affine intersections and affine-quadratic phases. For equal-weight superpositions, the phase optimization collapses, and the threshold is determined entirely by how many CWS words can lie in affine flats of each dimension. Thus a quantum optimization over stabilizer states becomes a classical affine-incidence problem. This reduction yields a fixed-parameter algorithm, an analytic lower bound for an infinite union-stabilizer family, and exact rational certificates for several standard nonadditive CWS examples. The framework also clarifies why exact enumeration fails for large structured families and identifies the remaining task as an affine-intersection problem. The result provides a geometric mechanism by which nonlinear CWS word sets generate certifiable magic.

发表机构

• Chung Yuan Christian University（中原大学）
• National Center for Theoretical Sciences（国家理论科学中心）
• National Yang Ming Chiao Tung University（国立阳明交通大学）

机构由 AI 辅助整理，请以论文原文为准。