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

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 $$ 0.231 \%$$ as the threshold of the tailored surface code, which is much better than $$ 0.21 \%$$ and $$ 0.209 \%$$ 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 $$ 0.264\%$$ , but the thresholds of the tailored surface code and XZZX code are $$ 0.296 \% $$ and $$ 0.328 \%$$ respectively.