Numerically optimized quantum error-correcting codes for a bosonic mode

Bosonic quantum error-correcting codes, such as cat and binomial codes, provide good performance against photon loss errors and are experimentally realizable e.g. with superconducting circuits. Motivated by this and aiming to broaden the spectrum of bosonic codes, we have numerically searched for bosonic single-mode codes. The codes are constructed from a finite superposition of Fock states. By utilizing numerical methods, the complex coefficients of the Fock states are chosen so that a code protects against L photon loss errors and minimizes the probability for the L+1:th error to occur. We have found codes that are protected up 9 photon loss errors. We present the structure and analyze the performance of these numerically optimized codes.