Fault-tolerant quantum computation in bosonic systems with ancillas

The bosonic system is a promising candidate for quantum computing due to the hardware efficiency of its corresponding quantum error correction (QEC) scheme. One practical way to achieve the universal control of a bosonic mode is to couple it to additional discrete-level (DV) ancillas for nonlinear operations. However, such ancillas are typically more prone to environmental noise, limiting the fidelity of ancilla-assisted quantum operations. To address this challenge, we develop a general framework for fault-tolerant quantum computation (FTQC) on a bosonic system with DV ancillas composed of fault-tolerant operations against noises from both parts. We define the fault-tolerance criteria in this hybrid system and consider employing both rotation-symmetric codes and the Gottesman-Kitaev-Preskill (GKP) code. Several FTQC toolkits are fitted into our protocol: we generalize the path-independent (PI) gate design such that up to an acceptable error, the gate implemented on the bosonic mode depends only on the initial and final state of the ancilla; we also embed the "flag qubit" QEC protocol as an autonomous error indicator to further improve the performance. Numeric simulations show suppressions of higher-order errors. Our general framework can guide the design of FTQC with noisy bosonic-ancilla hybrid systems.