[論文レビュー] Non-Abelian qLDPC: TQFT Formalism, Addressable Gauging Measurement and Application to Magic State Fountain on 2D Product Codes

Guanyu Zhu, Ryohei Kobayashi+1 more|arXiv (Cornell University)|Jan 11, 2026

Quantum Computing Algorithms and Architecture被引用数 0

ひとこと要約

The paper builds non-Abelian qLDPC codes on Poincaré CW complexes, develops a TQFT-based path integral formalism, and uses addressable gauging measurements to realize native non-Clifford gates and a magic-state fountain on 2D product codes with constant rate and √n distance.

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 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.

研究の動機と目的

• Motivate and construct non-Abelian qLDPC codes beyond manifolds using combinatorial TQFT on Poincaré CW complexes.
• Introduce a path integral formalism that yields topological invariants and guides fault-tolerant logical actions.
• Show how addressable gauging measurements of subcomplex/higher-form symmetries realize logical non-Clifford gates.
• Demonstrate a parallel magic-state fountain on 2D (thickened or skeleton) hypergraph-product codes with scalable resource counts.

提案手法

• Map skeleton classical LDPC codes to high-dimensional CW complexes and their deformations to Poincaré complexes.
• Construct twisted (non-Abelian) qLDPC codes as stabilizer codes with X-stabilizers dressed by CZ gates.
• Define a spacetime path integral using cup products on these complexes to obtain topological invariants.
• Realize encoding rates and distances for thickened 2D hypergraph-product codes (constant rate, distance Ω(√n)).
• Develop an addressable gauging measurement protocol for higher-form and subcomplex symmetries to implement logical CZ gates.
• Show that the gauging measurement protocol yields a magic-state fountain enabling parallel preparation of disjoint magic states on 2D product codes.

実験結果

リサーチクエスチョン

• RQ1 Can non-Abelian qLDPC codes be realized with a combinatorial TQFT defined on Poincaré CW complexes beyond manifolds?
• RQ2 How can spacetime path integrals formalize fault-tolerant logical actions for qLDPC codes?
• RQ3 Can addressable gauging measurements of higher-form or subcomplex symmetries enable native non-Clifford gates on 2D product codes?
• RQ4 What is the scalability (rate, distance, qubit count) of a magic-state fountain using 2D hypergraph-product codes?
• RQ5 Do these constructions yield non-Abelian fusion/braiding statistics and Borromean-ring-type braiding consistent with twisted gauge theories?

主な発見

• First non-Abelian qLDPC codes and corresponding TQFTs are constructed on general chain complexes and Poincaré CW complexes.
• A spacetime path integral for a twisted higher-form gauge theory is defined as a topological invariant on these complexes.
• The framework provides addressable transversal gates via higher-form or subcomplex symmetries, enabling non-Clifford operations within 2D codes.
• A magic-state fountain is realized on 2D hypergraph-product codes with Θ(√n) disjoint CZ magic states and distance Ω(√n) in O(d) rounds using n qubits.
• The protocol connects path-integral gauging to logical actions, enabling parallel preparation of resource states for universal computation.
• Non-Abelian fusion and Borromean-ring braiding statistics are derived, illustrating genuine non-Abelian topological order in twisted qLDPC codes.