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Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error-correcting codes^,. When manipulating the logical quantum states, it is imperative that errors caused by imperfect operations do not spread uncontrollably through the quantum register. This requires that all operations on the quantum register obey a fault-tolerant circuit design^{, –}, which, in general, increases the complexity of the implementation. Here we demonstrate a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. In particular, we make use of the recently introduced paradigm of flag fault tolerance, where the absence or presence of dangerous errors is heralded by the use of auxiliary flag qubits^{, , , –}. We perform a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code^,, and fault-tolerantly prepare …