Fault-tolerant Fusion-based Quantum Computing with the Four-legged Cat Code

With its ability to correct for single-photon loss, the 4-legged cat code stands out as a promising bosonic quantum memory, becoming the first code to surpass the quantum error correction breakeven point. We proposed a fault-tolerant protocol for Bell measurements (fusions) using this code. In this talk, we extend the framework with error-detected circuits for preparing 3-body GHZ states and CZ gates. These operations can be combined with fusions to concatenate the cat code with an XZZX cluster state. These operations are efficiently implemented using standard cQED tools: inter-cavity beamsplitter coupling, cavity displacements, cavity-transmon dispersive coupling, and transmon drives. Analytical results and pulse-level simulations demonstrate that these operations are, at worst, second-order sensitive to the dominant errors in cQED hardware, such as photon loss in the storage cavity, decoherence in the transmons, and undesired non-linearities, particularly cavity self-Kerr. Implementing the XZZX cluster state with the 4-legged code exploits the code's inherent first-order robustness, promising a suppression of physical qubit error rates beyond what is expected from unencoded physical qubits. Finally, we propose a fully planar implementation of the XZZX cluster state by efficiently reusing hardware.