量子纠错中的子系统对称性与分形子模型
Subsystem Symmetries and Fracton Models in Quantum Error Correction
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中文总结 AI 辅助
该研究探究经典伊辛模型与量子纠错的关联，确定棋盘格码最优阈值达0.107(3)，证明分形子码是高鲁棒性量子存储器的优良候选，凸显统计力学框架的应用价值。
中文摘要 AI 辅助
构建新型量子码并理解其错误恢复能力是实现稳健量子存储器的核心挑战。拓扑码因具备优异的纠错特性，且与多体物理中的物质相存在关联，成为极具潜力的候选方案。本论文通过子系统对称性、分形子拓扑序以及Kramers-Wannier型对偶性，探究经典伊辛模型与量子纠错之间的相互作用。我们研究了两个具有子系统对称性的三维经典自对偶伊辛模型：四面体伊辛模型与分形伊辛模型，考察它们的热行为、通过子系统对称性规范变换与分形子相的关联，以及所得分形子码的特性。利用统计力学映射，我们确定棋盘格码（Checkerboard code）的最优码容量阈值为0.107(3)，该值达到了CSS码的理论极限，是已知三维码中最高的最优错误阈值。我们将这一饱和现象与满足Kramers-Wannier型对偶性的经典自旋模型的广义熵关系关联起来，并论证该预测如何推广到编码率为零的CSS码，其X噪声与Z噪声模型映射为经典对偶自旋模型。这些发现确立了分形子码作为高鲁棒性量子存储器候选的地位，证明了统计力学框架及其对偶性预测在分析和构建稳健量子纠错码方面的有效性。
英文摘要
Constructing new quantum codes and understanding their error resilience are central challenges in the development of robust quantum memories. Topological codes are particularly promising due to their favorable error-correcting properties and their connections to phases of matter in many-body physics. In this thesis, we explore the interplay between classical Ising models and quantum error correction through subsystem symmetries, fracton topological order, and Kramers-Wannier-type duality. We study two three-dimensional classical self-dual Ising models with subsystem symmetries, the Tetrahedral Ising model and the Fractal Ising model, investigating their thermal behavior, their relation to fracton phases through subsystem-symmetry gauging, and the properties of the resulting fracton codes. Using a statistical-mechanical mapping, we determine the optimal code-capacity threshold of the Checkerboard code to be $0.107(3)$, which saturates the theoretical limit for CSS codes and represents the highest optimal error threshold among known three-dimensional codes. We relate this saturation to a generalized entropy relation for classical spin models satisfying a Kramers-Wannier-type duality, and argue how this prediction extends to CSS codes with zero encoding rate whose $X$- and $Z$-noise models map to classically dual spin models. These findings establish fracton codes as highly resilient candidates for quantum memories and demonstrate the power of the statistical-mechanical framework, together with its duality predictions, in analyzing and constructing robust quantum error-correcting codes.
