  We demonstrate laser-driven two-qubit and single-qubit logic gates with
fidelities 99.9(1)% and 99.9934(3)% respectively, significantly above the
approximately 99% minimum threshold level required for fault-tolerant quantum
computation, using qubits stored in hyperfine ground states of calcium-43 ions
held in a room-temperature trap. We study the speed/fidelity trade-off for the
two-qubit gate, for gate times between 3.8$\mu$s and 520$\mu$s, and develop a
theoretical error model which is consistent with the data and which allows us
to identify the principal technical sources of infidelity.
