  Quantum sensors can potentially achieve the Heisenberg limit of sensitivity
over a large dynamic range using quantum algorithms. The adaptive phase
estimation algorithm (PEA) is one example that was proven to achieve such high
sensitivities with single-shot readout (SSR) sensors. However, using the
adaptive PEA on a non-SSR sensor is not trivial due to the low contrast nature
of the measurement. The standard approach to account for the averaged nature of
the measurement in this PEA algorithm is to use a method based on `majority
voting'. Although it is easy to implement, this method is more prone to
mistakes due to noise in the measurement. To reduce these mistakes, a binomial
distribution technique from a batch selection was recently shown theoretically
to be superior, as all ranges of outcomes from an averaged measurement are
considered. Here we apply, for the first time, real-time non-adaptive PEA on a
non-SSR sensor with the binomial distribution approach. We compare the mean
square error of the binomial distribution method to the majority-voting
approach using the nitrogen-vacancy center in diamond at ambient conditions as
a non-SSR sensor. Our results suggest that the binomial distribution approach
achieves better accuracy with the same sensing times. To further shorten the
sensing time, we propose an adaptive algorithm that controls the readout phase
and, therefore, the measurement basis set. We show by numerical simulation that
adding the adaptive protocol can further improve the accuracy in a future
real-time experiment.
