Phase Error Model for Quantum Multiphase Estimation

The Quantum and Classical Cramér Rao Bound (QCRB and CCRB) set a lower bound for the total variance in quantum multiphase estimation. Although they are frequently used to show the potential of multiphase estimation to surpass the Heisenberg Limit (HL), an accurate model for total variance is needed to confirm the attainability of this advantage in practice. We derive a phase error model that evaluates the variance in multiphase estimation given a probe state and a measurement basis. We simulate the multiphase estimation using a generalized N00N state as the probe and various measurement bases, with results of numerical experiments compared to the proposed error model. A Chi-square test is performed to show that our phase error model is capable of accurately predicting the variance of phase estimation. We further exploit machine learning methods to minimize the error model and CCRB with respect to initial local phases in the generalized N00N state for a particular measurement basis. The scaling of the minimal values of the error model with the number of modes of the N00N state is analyzed and compared with that of the CCRB and QCRB. Our analysis confirms the potential of surpassing the HL using multiphase estimation methods in practice.