Optimality of Gottesman-Kitaev-Preskill (GKP) Codes for Bosonic Quantum Error Correction

Bosonic quantum error correction has recently risen as a hardware-efficient alternative to the conventional multi-qubit-based quantum error correction. We mainly focus on photon loss error, which is a dominant error source in microwave cavity modes. Previously, it was shown that GKP codes outperform many other bosonic quantum error-correcting codes in correcting photon loss errors, despite the fact that GKP codes are not designed to correct loss errors [1]. Here, we explain why GKP codes perform well against photon loss errors by providing a near-optimal decoding scheme and analyzing its performance. Furthermore, we formulate a biconvex optimization to find the best single-mode bosonic error-correcting code for photon loss errors. In particular, we solve the biconvex optimization heuristically by an alternating semi-definite programming method and show that, starting from Haar random initial codes, our numerical optimization yields a hexagonal GKP code as an optimal encoding in a practically relevant regime [2].

[1] V.V. Albert, et al., Phys. Rev. A 97, 032346 (2018),
[2] K. Noh, et al., arXiv:1801.07271 (2018): Accepted in IEEE Trans. Info. Theory.