High-fidelity initialization a logical qubit with multiple injections

Abstract: Quantum error correction represents a significant advancement in large-scale quantum computing. However, achieving fault-tolerant implementations of non-Clifford logical gates with reduced overhead remains a challenge in the popular surface code strategy. Recent advances have underscored the need for a substantial code distance to attain complete fault tolerance. Here, we introduce a continuous fault-tolerant scheme for non-Clifford logical gates via multiple injections. Unlike existing protocols that focus on a single logical chain, our approach utilizes multiple logical chains, each can employ the same or different logical rotation angles, to initialize a non-Clifford state. Compared to previous efforts, our protocol significantly alleviates the challenges associated with the requirement for a large code distance and reduces the corresponding resource overhead, making it more feasible to be implemented in current mid-scale chips via the surface code strategy.