Scalable hyperfine qubit state detection via electron shelving in the ${}^2$D$_{5/2}$ and ${}^2$F$_{7/2}$ manifolds in ${}^{171}$Yb$^{+}$

Qubits encoded in hyperfine states of trapped ions are ideal for quantum computation given their long lifetimes and low sensitivity to magnetic fields, yet they suffer from off-resonant scattering during detection often limiting their measurement fidelity. In ${}^{171}$Yb$^{+}$ this is exacerbated by a low fluorescence yield, which leads to a need for complex and expensive hardware - a problematic bottleneck especially when scaling up the number of qubits. We demonstrate a detection routine based on electron shelving to address this issue in ${}^{171}$Yb$^{+}$ and achieve a 5.6$times$ reduction in single-ion detection error on an avalanche photodiode to $1.8(2)times10^{-3}$ in a 100 $mu$s detection period, and a 4.3$times$ error reduction on an electron multiplying CCD camera, with $7.7(2)times10^{-3}$ error in 400 $mu$s. We further improve the characterization of a repump transition at 760 nm to enable a more rapid reset of the auxiliary $^2$F$_{7/2}$ states populated after shelving. Finally, we examine the detection fidelity limit using the long-lived $^2$F$_{7/2}$ state, achieving a further 300$times$ and 12$times$ reduction in error to $6(7)times10^{-6}$ and $6.3(3)times10^{-4}$ in 1 ms on the respective detectors. While shelving-rate limited in our setup, we suggest various techniques to realize this detection method at speeds compatible with quantum information processing, providing a pathway to ultra-high fidelity detection in ${}^{171}$Yb$^{+}$.