  Fault-tolerant quantum computing (FTQC) is essential for achieving
large-scale practical quantum computation. Implementing arbitrary FTQC requires
the execution of a universal gate set on logical qubits, which is highly
challenging. Particularly, in the superconducting system, two-qubit gates on
surface code logical qubits have not been realized. Here, we experimentally
implement logical CNOT gate as well as arbitrary single-qubit rotation gates on
distance-2 surface codes using the superconducting quantum processor
\textit{Wukong}, thereby demonstrating a universal logical gate set. In the
experiment, we design encoding circuits to prepare the required logical states,
where the fidelities of the fault-tolerantly prepared logical states surpass
those of the physical states. Furthermore, we demonstrate the transversal CNOT
gate between two logical qubits and fault-tolerantly prepare four logical Bell
states, all with fidelities exceeding those of the Bell states on the physical
qubits. Using the logical CNOT gate and an ancilla logical state, arbitrary
single-qubit rotation gate is implemented through gate teleportation. All
logical gates are characterized on a complete state set and their fidelities
are evaluated by logical Pauli transfer matrices. Implementation of the
universal logical gate set and entangled logical states beyond physical
fidelity marks a significant step towards FTQC on superconducting quantum
processors.
