Non-Abelian Quantum Low-Density Parity Check Codes and Non-Clifford Operations from Gauging Logical Gates via Measurements

TL;DR

This work develops a systematic, measurement-based framework to construct non-Abelian qLDPC codes by gauging transversal Clifford gates between copies of Abelian qLDPC codes. It identifies two complementary gauging paradigms—homological gauging, which relies on a Poincaré duality–like structure, and graph gauging, which uses an ancilla graph to gauge a global gate—and shows that the gauged codes exhibit 2D non-Abelian topological-order–like properties such as non-Abelian braiding in the absence of a fixed spatial manifold. The authors derive explicit stabilizer forms, ground-state structures, and logical operators for these non-Abelian codes, including D4 fracton models and non-Abelian bivariate bicycle codes, and demonstrate that their gauging procedures enable magic-state preparation for implementing non-Clifford operations on general qLDPC codes. They further analyze two concrete routes to non-Abelian behavior (homological and graph-based) and discuss the potential for fault-tolerant implementations, decoders, and experimental realizations with long-range connectivity. Overall, the paper broadens the landscape of qLDPC codes by embedding non-Abelian order into LDPC frameworks and offers practical pathways toward universal quantum computation using non-Abelian qLDPC codes.

Abstract

In this work, we introduce constructions for non-Abelian qLDPC codes obtained by gauging transversal Clifford gates using measurement and feedback. In particular, we identify two qualitatively different approaches to gauging qLDPC codes to obtain their non-Abelian counterparts. The first approach applies to codes that exhibit a generalized form of Poincaré duality and leads to a qLDPC non-Abelian Clifford stabilizer code, whose stabilizers are reminiscent of the action of a Type-III twisted quantum double. Our second approach applies to general qLDPC codes, and uses a graph of ancilla qubits which may be tailored to properties of the input codes to gauge a single transversal gate. For both constructions, the resulting gauged codes are shown to have properties analogous to 2D non-Abelian topological order -- e.g. the analog of a single anyon on a torus. We conclude by demonstrating that our gauging procedures enable magic state preparation via the measurement of logical Clifford gates. Consequently, our gauging constructions offer a protocol for performing non-Clifford operations on any qLDPC code.