Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes
This work addresses the challenge of reconciling connectivity and universality in two-dimensional architectures for fault-tolerant quantum computation with qLDPC codes. By generalizing Kitaev’s non-Abelian topological code to non-Abelian qLDPC codes, the authors construct a combinatorial topological quantum field theory based on Poincaré CW complexes and introduce a spacetime path integral formulation to enable addressable gauge measurements. The key innovation lies in the first realization of native non-Clifford logical gates on constant-rate two-dimensional hypergraph product codes, achieved through an addressable measurement scheme rooted in 0-form subcomplex symmetries, which is further extended to higher-dimensional and higher-order symmetries. This approach is successfully applied to magic state distillation, enabling the parallel preparation of $O(\sqrt{n})$ disjoint CZ magic states, each with code distance $O(\sqrt{n})$, on $n$ physical qubits.