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

TL;DR

This paper introduces methods to construct non-Abelian quantum LDPC codes by gauging transversal Clifford gates, enabling non-Clifford operations and magic state preparation through measurement-based procedures.

Contribution

It presents two novel approaches to gauge qLDPC codes into non-Abelian forms, facilitating non-Clifford gates and topological order analogs in quantum error correction.

Findings

Gauged codes exhibit properties similar to 2D non-Abelian topological order.

Protocols for magic state preparation via measurement of logical Clifford gates.

Construction methods applicable to a broad class of 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 Poincar\'e 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.