Gate Set and Error Correction Protocol for the CLY Bosonic Code in Circuit QED

The Chuang-Leung-Yamamoto (CLY) bosonic code [1] enables the direct detection and correction of single photon loss errors, in contrast to the dual-rail code [2], which can only detect them. Each CLY codeword is an eigenstate of the no-jump operator, mitigating no-jump backaction relative to the single-mode binomial code [3].

A five-gate sequence consisting of three balanced beamsplitter gates and two SNAP gates allows for single-qubit universal control within the CLY code. We show that this sequence is robust against single-photon loss anywhere in the five-gate sequence and that error correction can be delayed until the end due to the sequence's "error closure” property. We present an error-correction protocol for the CLY code using parity projectors, selective photon addition, and eSWAP gates that returns the system to the code space following single-photon loss errors. Finally, we show analytical and numerical results for a fault tolerant implementation of this protocol.



[1] I.L. Chuang et al., Phys. Rev. A, 56, 2 (1997).

[2] J. D. Teoh et al., Proc. Natl. Acad. Sci., 120, 41 (2023).

[3] M.H. Michael et al., Phys. Rev. X, 6, 3 (2016).