  We propose and demonstrate a protocol for high-fidelity indirect readout of
trapped ion hyperfine qubits, where the state of a $^9\text{Be}^+$ qubit ion is
mapped to a $^{25}\text{Mg}^+$ readout ion using laser-driven Raman
transitions. By partitioning the $^9\text{Be}^+$ ground state hyperfine
manifold into two subspaces representing the two qubit states and choosing
appropriate laser parameters, the protocol can be made robust to spontaneous
photon scattering errors on the Raman transitions, enabling repetition for
increased readout fidelity. We demonstrate combined readout and back-action
errors for the two subspaces of $1.2^{+1.1}_{-0.6} \times 10^{-4}$ and
$0^{+1.9}_{-0} \times 10^{-5}$ with 68% confidence while avoiding decoherence
of spectator qubits due to stray resonant light that is inherent to direct
fluorescence detection.

Quantum Physics;Atomic Physics