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

TL;DR

This paper extends non-Abelian topological codes to qLDPC codes, enabling native non-Clifford gates via a TQFT framework, and applies this to a magic state preparation scheme on 2D codes.

Contribution

It introduces a TQFT formalism for non-Abelian qLDPC codes and demonstrates how to implement non-Clifford gates and magic state distillation within this framework.

Findings

Native non-Clifford gates realized on 2D hypergraph-product codes.

Gauging measurement protocol enables parallel magic state preparation.

Constant-rate 2D codes achieve $O( oot{n})$ magic states with $n$ qubits.

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.