Continuous-variable Quantum Phase Estimation based on Machine Learning

Tailong Xiao,Jingzheng Huang,Guihua Zeng,Jianping Fan

doi:10.1038/s41598-019-48551-0

Tailong Xiao, Jingzheng Huang + Show 2 more

Open Access

https://doi.org/10.1038/s41598-019-48551-0

Copy DOI

Abstract

Making use of the general physical model of the Mach-Zehnder interferometer with photon loss which is a fundamental physical issue, we investigate the continuous-variable quantum phase estimation based on machine learning approach, and an efficient recursive Bayesian estimation algorithm for Gaussian states phase estimation has been proposed. With the proposed algorithm, the performance of the phase estimation may be improved distinguishably. For example, the physical limits (i.e., the standard quantum limit and Heisenberg limit) for the phase estimation precision may be reached in more efficient ways especially in the situation of the prior information being employed, the range for the estimated phase parameter can be extended from [0, π/2] to [0, 2π] compared with the conventional approach, and influences of the photon losses on the output parameter estimation precision may be suppressed dramatically in terms of saturating the lossy bound. In addition, the proposed algorithm can be extended to the time-variable or multi-parameter estimation framework.

Highlights

One class of quantum states of particular interest is the Gaussian states[23], which is a kind of continuous-variable (CV) quantum states since they are easy to produce and comparatively robust against losses
When |α|2 = 100, r = 0.8 and the phase shift is set to be optimal, we find that homodyne detectors (HD) with maximum likelihood estimation (MLE) cannot saturate sub-shot noise limit in a few trials (M < 300)
When r increases, we find that the gap between the sub-shot noise limit and the simulation results of HD with MLE becomes larger in the early stage of training

Summary

Phase Estimation based on Machine Learning

Making use of the general physical model of the Mach-Zehnder interferometer with photon loss which is a fundamental physical issue, we investigate the continuous-variable quantum phase estimation based on machine learning approach, and an efficient recursive Bayesian estimation algorithm for Gaussian states phase estimation has been proposed. Photon number detection can avoid the dependence of phase estimation accuracy on the true phase value because it fully utilizes the statistical feature of each photon These measurement methods for Heisenberg scaling are not implemented readily in practical quantum physical systems and more significantly, they are more sensitive to photon loss in lossy MZI46,47. When r increases, we find that the gap between the sub-shot noise limit and the simulation results of HD with MLE becomes larger in the early stage of training It indicates that MLE needs more samples to calculate the likelihood function and estimate the optimal phase parameter. We find that the Gaussian prior knowledge can certainly enhance the precision for coherent and CSV state with small measurement trials (M < 100), but its impact vanishes when trial becomes large because our algorithm uses the history measurement information to update the prior distribution, which validates the above theoretical analysis.

Fitted curve

Conclusions

Author Contributions

Additional Information