Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes

2026-01-11

AI Summary

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.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

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 Poincar\'e 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.

Problem

• qLDPC codes
• non-Clifford gates
• fault-tolerant quantum computation
• 2D product codes
• universality

Innovation

• non-Abelian qLDPC
• topological quantum field theory
• addressable gauging measurement
• 2D hypergraph-product codes
• magic state fountain