Poking holes and cutting corners to achieve Clifford gates with the surface code

TL;DR

This paper unifies various methods for implementing fault-tolerant Clifford gates in the surface code, introducing new schemes and demonstrating all Clifford gates can be achieved via code deformation without reducing code distance.

Contribution

It provides a unified framework for different Clifford gate implementations in the surface code and introduces a new logical encoding and hybrid schemes.

Findings

All Clifford gates can be implemented with the planar code via code deformation.

Identification of twist defects with code corners enables new gate operations.

Proposes a method to perform gates without loss of code distance.

Abstract

The surface code is currently the leading proposal to achieve fault-tolerant quantum computation. Among its strengths are the plethora of known ways in which fault-tolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes and even by braiding engineered Majorana modes using twist defects. Here we present a unified framework to describe these methods, which can be used to better compare different schemes, and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform single-qubit Clifford gates by exchanging the corners of the planar code via code deformation. We analyse ways in which different schemes can be combined, and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery.