Logical coherence in 2D compass codes

TL;DR

This paper analyzes the logical coherence thresholds of 2D compass codes, providing new analytical bounds and demonstrating their potential to improve understanding of quantum error correction under coherent rotations.

Contribution

It introduces two new compass code families with analytically determined thresholds and uses simulation to explore code interpolations between known codes.

Findings

Analytical thresholds for new compass code families are established.

The $Z$-Shor code's lower bound on the threshold matches the achievable bound.

Simulations show smooth interpolation between $Z$-Shor and $X$-Shor codes.

Abstract

2D compass codes are a family of quantum error-correcting codes that contain the Bacon-Shor codes, the $X$-Shor and $Z$-Shor codes, and the rotated surface codes. Previous numerical results suggest that the surface code has a constant accuracy and coherence threshold under uniform coherent rotation. However, having analytical proof supporting a constant threshold is still an open problem. It is analytically proven that the toric code can exponentially suppress logical coherence in the code distance $L$. However, the current analytical lower bound on the threshold for the rotation angle $\theta$ is $|\sin(\theta)| < 1/L$, which linearly vanishes in $L$ instead of being constant. We show that this lower bound is achievable by the $Z$-Shor code which does not have a threshold under stochastic noise. Compass codes provide a promising direction to improve on the previous bounds. We analytically determine thresholds for two new compass code families that provide upper and lower bounds to the rotated surface code's numerically established infidelity threshold. Furthermore, using a Majorana mode-based simulator, we use random families of compass codes to smoothly interpolate between the $Z$-Shor codes and the $X$-Shor codes.