  We introduce a novel Bayesian phase estimation technique based on adaptive
grid refinement method. This method automatically chooses the number particles
needed for accurate phase estimation using grid refinement and cell merging
strategies such that the total number of particles needed at each step is
minimal. The proposed method provides a powerful alternative to traditional
sampling based sequential Monte Carlo method which tend to fail in certain
instances such as when the posterior distribution is bimodal. We also combine
grid based and sampling based methods as hybrid particle filter where grid
based method can be used to estimate a small but dominant set of parameters and
Liu-West (LW) based SMC for the remaining set of parameters. Principal kurtosis
analysis can be used to decide the choice of parameters for grid refinement
method and for sampling based methods. We provide numerical results comparing
the performance of the proposed grid refinement method with Liu-West resampling
based SMC. Numerical results suggest that the proposed method is quite
promising for quantum phase estimation. It can be easily adapted to Hamiltonian
learning which is a very useful technique for estimating unknown parameters of
a Hamiltonian and for characterizing unknown quantum devices.
