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.

中文翻译：

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具有自适应网格细化的贝叶斯相位估计

我们介绍了一种基于自适应网格细化方法的新型贝叶斯相位估计技术。该方法使用网格细化和单元合并策略自动选择精确相位估计所需的粒子数，从而使每一步所需的粒子总数最小。所提出的方法为传统的基于采样的顺序蒙特卡罗方法提供了一种强大的替代方法，该方法在某些情况下往往会失败，例如当后验分布是双峰时。我们还将基于网格和基于采样的方法组合为混合粒子滤波器，其中基于网格的方法可用于估计少量但占主导地位的参数集，并基于 Liu-West (LW) 的 SMC 来估计剩余的参数集。主峰态分析可用于决定网格细化方法和基于采样的方法的参数选择。我们提供了数值结果，比较了所提出的网格细化方法与基于 Liu-West 重采样的 SMC 的性能。数值结果表明，所提出的方法对于量子相位估计很有前景。它可以很容易地适应哈密顿学习，这是一种非常有用的技术，用于估计哈密顿量的未知参数和表征未知量子设备。