A fundamental problem of fault-tolerant quantum computation with quantum low-density parity-check (qLDPC) codes is the tradeoff between connectivity and universality. It is widely believed that in order to perform native logical non-Clifford gates, one needs to resort to 3D product-code constructions. In this work, we extend Kitaev's framework of non-Abelian topological codes on manifolds to non-Abelian qLDPC codes (realized as Clifford-stabilizer codes) and the corresponding combinatorial topological quantum field theories (TQFT) defined on Poincaré CW complexes and certain types of general chain complexes. We also construct the spacetime path integrals as topological invariants on these complexes. Remarkably, we show that native non-Clifford logical gates can be realized using constant-rate 2D hypergraph-product codes and their Clifford-stabilizer variants. This is achieved by a spacetime path integral effectively implementing the addressable gauging measurement of a new type of 0-form subcomplex symmetries, which correspond to addressable transversal Clifford gates and become higher-form symmetries when lifted to higher-dimensional CW complexes or manifolds. Building on this structure, we apply the gauging protocol to the magic state fountain scheme for parallel preparation of $O(\sqrt{n})$ disjoint CZ magic states with code distance of $O(\sqrt{n})$, using a total number of $n$ qubits.

翻译：容错量子计算中量子低密度奇偶校验码面临的一个基本问题是连通性与通用性之间的权衡。学界普遍认为，为实现本征逻辑非克利福德门，必须采用三维乘积码构造。本研究中，我们将Kitaev关于流形上非阿贝尔拓扑码的框架扩展至非阿贝尔量子低密度奇偶校验码（实现为克利福德稳定子码）及其对应的组合拓扑量子场论——这些理论定义在庞加莱CW复形与特定类型的广义链复形上。同时，我们在这些复形上构建了作为拓扑不变量的时空路径积分。值得注意的是，我们证明了本征非克利福德逻辑门可通过恒定速率的二维超图乘积码及其克利福德稳定子变体实现。这是通过时空路径积分有效实现新型零形式子复形对称性的可寻址规范测量达成的，这些对称性对应于可寻址横截克利福德门，当提升至高维CW复形或流形时则成为高形式对称性。基于此结构，我们将规范协议应用于魔态喷泉方案，通过总计$n$个量子比特，实现了$O(\sqrt{n})$个互不相交的CZ魔态并行制备，其码距为$O(\sqrt{n})$。