  Using two trapped ion species ($\rm{^{27}Al^+}$ and $\rm{^9Be^+}$) as primary
and ancillary systems, we implement qubit measurements based on the repetitive
transfer of information and quantum nondemolition detection. The repetition
provides a natural mechanism for an adaptive measurement strategy, which leads
to exponentially lower error rates compared to using a fixed number of
detection cycles. For a single qubit we demonstrate 99.94 % measurement
fidelity. We also demonstrate a technique for adaptively measuring multiple
qubit states using a single ancilla, and apply the technique to spectroscopy of
an optical clock transition.
