Design of Quantum error correcting code for biased error on heavy-hexagon structure

TL;DR

This paper develops methods to implement tailored surface and XZZX quantum error-correcting codes on heavy-hexagon structures, showing improved error thresholds especially under biased error conditions in transmon-based quantum computers.

Contribution

It provides a novel implementation approach for tailored surface and XZZX codes on heavy-hexagon structures, with enhanced error thresholds for biased errors.

Findings

Tailored surface code threshold is 0.231779% without bias.

Thresholds increase with Z error bias for both codes.

In infinite bias, tailored codes outperform standard surface code thresholds.

Abstract

Surface code is an error-correcting method that can be applied to the implementation of a usable quantum computer. At present, a promising candidate for a usable quantum computer is based on superconductor-specifically transmon. Because errors in transmon-based quantum computers appear biasedly as Z type errors, tailored surface and XZZX codes have been developed to deal with the type errors. Even though these surface codes have been suggested for lattice structures, since transmons-based quantum computers, developed by IBM, have a heavy-hexagon structure, it is natural to ask how tailored surface code and XZZX code can be implemented on the heavy-hexagon structure. In this study, we provide a method for implementing tailored surface code and XZZX code on a heavy-hexagon structure. Even when there is no bias, we obtain $ 0.231779 \%$ as the threshold of the tailored surface code, which is much better than $ 0.210064 \%$ and $ 0.209214 \%$ as the thresholds of the surface code and XZZX code, respectively. Furthermore, we can see that even though a decoder, which is not the best of the syndromes, is used, the thresholds of the tailored surface code and XZZX code increase as the bias of the Z error increases. Finally, we show that in the case of infinite bias, the threshold of the surface code is $ 0.264852\%$, but the thresholds of the tailored surface code and XZZX code are $ 0.296157 \% $ and $ 0.328127 \%$ respectively.