Fault-tolerant error correction in CLY code

Bosonic codes stored in superconducting microwave cavities are an excellent candidate for quantum computing. They offer the advantage of a longer lifetime compared to circuit qubits and a larger Hilbert space that allows more complicated qubit encodings with error detection and correction. We will focus on the Chuang-Leung-Yamamoto (CLY) code which uses the dual-rail setup with multi-photon logical states. The CLY code allows for the detection and correction of the loss of a single cavity photon. The dominant error during the error-correction protocol is the decay of the auxiliary transmon qubit. For fault-tolerance error correction, it is necessary to use the transmon states g and f as the logical states of the ancilla and make transmon measurements during the protocol. The transmon measurements themselves add error which leads to a complicated decision tree regarding the optimal measurement points and error-correction sequences. We examine the optimal error-correction setup and its dependence on various system parameters such as the intrinsic lifetimes of the different components and the coupling strengths between them.