Single-mode bosonic error correcting codes with rotation symmetry

Bosonic mode error correcting codes, or mode codes for short, are error correcting codes where a qubit (or qudit) is encoded into one or multiple bosonic modes, i.e., quantum oscillators with an infinite Hilbert space. In this talk I focus on single-mode codes that obey rotation symmetry in phase space, such as the the well known Cat-, Binomial- and GKP codes. I will introduce a universal scheme for this class of codes based only on simple and experimentally well-motivated interactions. The scheme is fault-tolerant in the sense that small errors are guaranteed to remain small under the considered gates. I will also introduce a fault-tolerant error correction scheme based on cross-Kerr interactions and destructive phase measurement (e.g., heterodyne). Remarkably, the error correction scheme approaches the optimal recovery map for Cat and Binomial codes when the ancilla modes are error free. We numerically compute break-even thresholds under loss and dephasing, both with ideal and faulty ancillas.