Surface code dislocations have code distance L+O(1)

TL;DR

This paper revises the understanding of surface code dislocations' code distance, demonstrating that allowing Y errors reduces the code distance from 2L + O(1) to L + O(1).

Contribution

It corrects prior assumptions by showing the impact of Y errors on the code distance of surface code dislocations.

Findings

Code distance reduces from 2L + O(1) to L + O(1) with Y errors.

Y errors significantly affect the robustness of surface code dislocations.

The result impacts fault-tolerance thresholds for surface codes.

Abstract

In [Hastings et al 2014] it is stated that the code distance of a logical qubit stored using dislocations is 2L + O(1), where L is the separation between the dislocation twists. This code distance assumed only physical X and Z errors are permitted. This short note shows that, when Y errors are allowed, the code distance reduces to L + O(1). See Figure 1.