We demonstrate single-shot qubit readout with fidelity sufficient for fault-tolerant quantum computation, for two types of qubit stored in single trapped calcium ions. For an optical qubit stored in the (4S_1/2, 3D_5/2) levels of 40Ca+ we achieve 99.991(1)% average readout fidelity in one million trials, using time-resolved photon counting. An adaptive measurement technique allows 99.99% fidelity to be reached in 145us average detection time. For a hyperfine qubit stored in the long-lived 4S_1/2 (F=3, F=4) sub-levels of 43Ca+ we propose and implement a simple and robust optical pumping scheme to transfer the hyperfine qubit to the optical qubit, capable of a theoretical fidelity 99.95% in 10us. Experimentally we achieve 99.77(3)% net readout fidelity, inferring at least 99.87(4)% fidelity for the transfer operation.
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