  In recent years there has been substantial development in algorithms for
quantum phase estimation. In this work we provide a new approach to online
Bayesian phase estimation that achieves Heisenberg limited scaling that
requires exponentially less classical processing time with the desired error
tolerance than existing Bayesian methods.
  This practically means that we can perform an update in microseconds on a CPU
as opposed to milliseconds for existing particle filter methods. Our approach
assumes that the prior distribution is Gaussian and exploits the fact, when
optimal experiments are chosen, the mean of the prior distribution is given by
the position of a random walker whose moves are dictated by the measurement
outcomes. We then argue from arguments based on the Fisher information that our
algorithm provides a near-optimal analysis of the data. This work shows that
online Bayesian inference is practical, efficient and ready for deployment in
modern FPGA driven adaptive experiments.
