Quantum error correction of a multimode grid code using confidence information, Part 1: Theory

By encoding the logical information in the large Hilbert space of harmonic oscillators, bosonic codes are a promising approach to hardware-efficient quantum computing. The single-mode Gottesman-Kitaev-Preskill (GKP) code [1], in particular, has been shown theoretically to outperform other single-mode bosonic codes against photon loss under an ideal recovery channel [2, 3]. However, the practical quantum error correction (QEC) of single-mode GKP codes is limited by the error rates of the auxiliary elements needed to control the bosonic mode.

Here, we investigate the tesseract code, a two-mode grid code that has been proposed to be more robust to auxiliary errors than the single-mode codes [4]. Using analytical and numerical tools, we show that a single decay error of the auxiliary transmon cannot cause undetectable logical errors in the tesseract code. Finally, we discuss how the detection of errors can be used as real-time confidence information during the QEC of tesseract qubits.

[1] D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)

[2] V. V. Albert et al., Phys. Rev. A 97, 032346 (2018)

[3] G. Zheng et al., arXiv:2401.02022 (2024)

[4] B. Royer, S. Singh, and S. M. Girvin, PRX Quantum 3, 010335 (2022)