Non-Clifford gates between stabilizer codes via non-Abelian topological order

TL;DR

This paper introduces protocols for implementing non-Clifford logical gates between stabilizer codes by leveraging non-Abelian topological order, expanding the computational capabilities of quantum codes beyond traditional methods.

Contribution

It provides a general framework for generating non-Clifford gates between qudit surface codes using non-Abelian topological phases, with detailed protocols involving the quantum double of S3.

Findings

Demonstrated a protocol for a controlled-charge conjugation gate between qubit and qutrit surface codes.

Outlined fault-tolerance considerations and heralded decoding strategies.

Showed how to extend protocols to other non-Abelian groups for logical gate implementation.

Abstract

We propose protocols to implement non-Clifford logical gates between stabilizer codes by entangling into a non-Abelian topological order as an intermediate step. Generalizing previous approaches, we provide a framework that generates a large class of non-Clifford and non-diagonal logical gates between qudit surface codes by gauging the topological symmetry of symmetry-enriched topological orders. As our main example, we concretely detail a protocol that utilizes the quantum double of $S_3$ to generate a controlled-charge conjugation ($C\mathcal{C}$) gate between a qubit and qutrit surface code. Both the preparation of non-Abelian states and logical state injection between the Abelian and non-Abelian codes are executed via finite-depth quantum circuits with measurement and feedforward. We discuss aspects of the fault-tolerance of our protocol, presenting insights on how to construct a heralded decoder for the quantum double of $S_3.$ We also outline how analogous protocols can be used to obtain logical gates between qudit surface codes by entangling into $\mathcal{D}(G),$ where $G$ is a semidirect product of Abelian groups. This work serves as a step towards classifying the computational power of non-Abelian quantum phases beyond the paradigm of anyon braiding on near-term quantum devices.