Hybrid Lattice Surgery: Non-Clifford Gates via Non-Abelian Surface Codes

TL;DR

This paper introduces hybrid lattice surgery, a resource-efficient method for implementing non-Clifford gates in surface codes by combining Abelian and non-Abelian topological codes, enabling universal fault-tolerant quantum computing.

Contribution

It generalizes lattice surgery to hybrid operations across different topological codes, facilitating non-Clifford gates in surface codes with a topological field theory framework.

Findings

Enables non-Clifford gates via hybrid lattice surgery.

Provides a topological field theory description of the process.

Extends protocols to higher Clifford hierarchy levels and qutrits.

Abstract

In universal fault-tolerant quantum computing, implementing logical non-Clifford gates often demands substantial spacetime resources for many error-correcting codes, including the high-threshold surface code. A critical mission for realizing large-scale quantum computing is to develop simple and resource-efficient implementations of logical non-Clifford gates. We propose a novel way of implementing non-Clifford operations in the standard surface code based on hybrid lattice surgery. First we generalize the standard lattice surgery to hybrid lattice surgery, where operations of rough merge and rough split happen across different topological codes. Then we apply such procedures between Abelian and non-Abelian codes and show that this can provide non-Clifford operations in the standard surface code, in the form of a magic state or a non-Clifford gate teleportation. Complementing this, we provide a continuum topological field theory description of this hybrid lattice surgery utilizing interfaces between (2+1)d topological orders. From these considerations, we can generalize our protocol to non-Clifford gates and magic states at all finite levels of the Clifford hierarchy, as well as gates beyond the hierarchy. We also discuss protocols extending this framework to qutrits.