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

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.

18 pages, 15 figures, comments welcome; v2 - improvements and clarifications given in the manuscript. No significant changes in the main conclusions; v3 - same as v2 with corrected fig. 2 and ; v4 - minor changes following peer review. One new figure; v5 - version to be published, final author copy, minor typos and mistakes corrected

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