Abstract
Bayesian estimation is a powerful theoretical paradigm for the operation of the approach to parameter estimation. However, the Bayesian method for statistical inference generally suffers from demanding calibration requirements that have so far restricted its use to systems that can be explicitly modeled. In this theoretical study, we formulate parameter estimation as a classification task and use artificial neural networks to efficiently perform Bayesian estimation. We show that the network’s posterior distribution is centered at the true (unknown) value of the parameter within an uncertainty given by the inverse Fisher information, representing the ultimate sensitivity limit for the given apparatus. When only a limited number of calibration measurements are available, our machine-learning-based procedure outperforms standard calibration methods. Our machine-learning-based procedure is model independent, and is thus well suited to “black-box sensors”, which lack simple explicit fitting models. Thus, our work paves the way for Bayesian quantum sensors that can take advantage of complex nonclassical quantum states and/or adaptive protocols. These capabilities can significantly enhance the sensitivity of future devices.
Highlights
Precise parameter estimation in quantum systems can revolutionize current technology and prompt scientific discoveries[1,2]
We propose that parameter estimation can be formulated as a classification task—similar to the identification of handwritten digits, see Fig. 1—able to be performed efficiently with supervised learning techniques based on artificial neural networks[34,35,36]
We show that our Bayesian parameter estimation (BPE) protocol is asymptotically unbiased and consistent: it obeys relevant Bayesian bounds[17] dictated, in our examples, by quantum and statistical noise
Summary
Precise parameter estimation in quantum systems can revolutionize current technology and prompt scientific discoveries[1,2]. To employ BPE in systems that cannot be modeled, methods must be developed to efficiently calibrate the device given limited data In this manuscript, we provide a machine-learning approach to BPE. We design a neural network adapted for parameter estimation whose output is, naturally, a Bayesian parameter distribution Based on this interpretation, we provide a theoretical framework that enables a network to be trained using the outcome of individual measurement results. The machinelearning-based parameter estimation illustrated in this manuscript can be readily applied for data analysis in current quantum sensors, providing all the important advantages of BPE, while enjoying less stringent calibration/training requirements. Noise and decoherence that affect the apparatus are directly included (via the training process) in the Bayesian posterior distributions which fully account for experimental imperfections
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