We report high-fidelity laser-beam-induced quantum logic gates on magnetic-field-insensitive qubits comprised of hyperfine states in $^{9}$Be$^+$ ions with a memory coherence time of more than 1 s. We demonstrate single-qubit gates with error per gate of $3.8(1)times 10^{-5}$. By creating a Bell state with a deterministic two-qubit gate, we deduce a gate error of $8(4)times10^{-4}$. We characterize the errors in our implementation and discuss methods to further reduce imperfections towards values that are compatible with fault-tolerant processing at realistic overhead.
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