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

The four-legged cat code is a quantum error-correcting code designed to address the predominant error in bosonic modes: single-photon loss. It was the first such code to surpass the break-even point, thereby demonstrating the practical utility of quantum error correction. In this work, we propose a planar fault-tolerant architecture for this code by concatenating it with the XZZX code via fusion-based error-correction. To the best of our knowledge, this is the first 2D nearest-neighbor architecture for fault-tolerant fusion-based error-correction. We demonstrate how all the required operations, namely resource state preparation and Bell measurements, can be carried out using standard circuit-QED techniques, such as intercavity beam-splitter coupling, cavity displacements, cavity-transmon dispersive coupling, and transmon drives. We show analytically and numerically that all dominant hardware errors in the bosonic modes and control ancillae are corrected, to first-order, at the hardware level. Consequently, the outer XZZX code only needs to address smaller residual errors, which are quadratically suppressed, effectively doubling the architecture's fault-distance. Moreover, the performance of our architecture is not limited by unwanted nonlinearities such as cavity self-Kerr, and it avoids demanding coupling techniques like $\chi$-matching or high-order coupling. Overall, our architecture substantially reduces the hardware complexity needed to achieve fault tolerance with the four-legged cat code.