Universal logical operations in a silicon quantum processor.

Quantum errors induced by environmental noise are unavoidable and preclude the direct implementation of practical quantum computation. Fault-tolerant quantum computation offers one of the viable paths, necessitating the encoding and processing of information within logical qubits to curb such errors. Although substantial progress has been achieved recently in building silicon quantum computers, logical operations still haven't been realized in silicon. Here we demonstrate a logical quantum processor using a phosphorus donor cluster in silicon. By implementing the [[4, 2, 2]] code, we realize the essential components for logical operations, which include fault-tolerant preparation of logical states and the characterization of a universal gate set comprising logical single-qubit and two-qubit gates. In particular, the logical T gate is achieved using the gate-by-measurement method, and magic states based on this gate are prepared. Furthermore, we execute the variational quantum eigensolver algorithm using two logical qubits and simulate the ground state of the electronic structure of the water molecule HO. This work represents a key step towards scalable, fault-tolerant quantum computation in silicon spin qubits.