AI 中文总结
本研究引入耦合层码统一量子乘积码与物质相的耦合层构造，可生成非CSS码，还能重现X立方等模型及费米子环面码等三维模型，拓展了量子纠错码的构造方法。
AI 中文摘要
乘积码是一类由多个输入码构造的重要量子纠错码，可生成渐近良好的量子低密度奇偶校验码。此前研究中，我们证明了两个码之间的乘积可通过利用第二个码的校验项耦合第一个码的层来实现。本研究通过引入耦合层码，进一步将乘积码构造与物质相的耦合层构造统一起来。核心策略是凝聚第一个码的多个解耦层中由泡利算子产生的一般激发，该凝聚由激发代数指定，激发代数编码激发及保代数映射。这种层间耦合推广了物理学中的规范概念以及同调代数中的映射锥，可用于生成非CSS码。作为示例，我们展示耦合层码如何重现X立方模型和Chamon模型。我们进一步推广平衡乘积码，除自由置换的群作用外还允许幺正变换，并描述其对应的耦合层构造。特别地，我们证明通过ZX对偶性平衡可重现非CSS码，如费米子环面码和三维3费米子Walker-Wang模型。
英文摘要
Product codes are an important class of quantum error-correcting codes constructed from multiple input codes, which can give rise to asymptotically good quantum low-density parity check codes. In previous work, we showed how the product between two codes can be physically implemented by coupling layers of the first code using checks of the second code. In this work, we further unify product code constructions with coupled-layer constructions of phases of matter by introducing coupled-layer codes. The essential strategy is to condense general excitations created by Pauli operators among multiple decoupled layers of the first code. The condensation is specified by an excitation algebra, which encodes the excitations, along with an algebra-preserving map. This coupling between layers generalizes the notion of gauging in physics as well as the mapping cone in homological algebra, and can be used to produce non-CSS codes. As examples, we show how coupled-layer codes reproduce the X-cube and Chamon models. We further generalize the balanced product code by allowing a unitary transformation in addition to a group action by free permutation and describe its corresponding coupled-layer construction. In particular, we show how balancing by a ZX-duality can reproduce non-CSS codes such as the fermionic toric code and the 3-fermion Walker-Wang model in 3D.
Comments27+15 pages
