Quantum error correction in cat-code qubits: analysis and minimization of failure rates

A bosonic mode coupled to an ancillary nonlinear circuit can be used to encode a logical qubit. One choice studied in the context of quantum error correction is the cat code. Repeated measurement of photon-number parity was shown by Ofek et al. to protect qubit from the dominant error (single-photon loss). However, a number of undetectable errors and failure modes remain. For cat qubits using either a transmon or fluxonium as the ancilla, we survey and categorize the major failure modes, providing efficient methods to estimate failure rates. By minimizing the total failure rate, we obtain optimal parameters for both the physical system and the protocol.