Circuit-level fault tolerance of cat codes

Bosonic codes encode quantum information into a single infinite-dimensional physical system endowed with error correction capabilities. This reduces the need for complex management of many physical constituents compared with standard approaches employing multiple physical qubits. Recent discussions of bosonic codes centre around correcting only boson-loss errors, with phase errors either actively suppressed or deferred to subsequent layers of encoding with standard qubit codes. Rotationally symmetric bosonic (RSB) codes, which include the well-known cat and binomial codes, are capable of simultaneous correction of loss and phase errors, offering an alternate route that deals with arbitrary errors already at the base layer. Here, we investigate the robustness of such codes, moving away from the more idealistic past studies towards a circuit-level noise analysis closer to the practical situation where every physical component in the device is potentially faulty. We extend the concept of fault tolerance to the case of RSB codes, and then examine the performance of two known error correction circuits under circuit-level noise. Our analysis reveals a significantly more stringent noise threshold for fault-tolerant operation than found in past works; nevertheless, we show how, through waiting-time optimization and the use of squeezing, we can restore the noise requirements to a regime achievable with near-term quantum hardware. While our focus here is on cat codes for concreteness, a similar analysis applies for general RSB codes.