Calibration of Quantum Sensors by Neural Networks

Introducing quantum sensors as a solution to real world problems demands reliability and controllability outside of laboratory conditions. Producers and operators ought to be assumed to have limited resources readily available for calibration, and yet, they should be able to trust the devices. Neural networks are almost ubiquitous for similar tasks for classical sensors: here we show the applications of this technique to calibrating a quantum photonic sensor. This is based on a set of training data, collected only relying on the available probe states, hence reducing overhead. We found that covering finely the parameter space is key to achieving uncertainties close to their ultimate level. This technique has the potential to become the standard approach to calibrate quantum sensors.

Figure 1

Schematics of the use of neural networks for parameter estimation. The first step (a) consists of training the network by inputting a set of training data using bootstrapping to account for uncertainties. Upon completion, the actual estimation (b) uses the trained network to extract an estimate of the parameter.

Figure 2

(a) Experimental setup. The $N00N$ state probe accumulates a phase $\varphi =4\chi$ by means of a test half wave plate ($t$-HWP). The analysis is carried out by a second measurement plate ($m$-HWP), a polarizing beam splitter, and coincidence detection of two avalanche photodiodes (APDs). (b) Part of the experimental training set. The registered count rates and those derived from bootstrapping are reported as a function of the calibrated phases $\varphi$. The labels indicate the angular setting $\theta$ of the measurement HWP, corresponding to four different polarization projections.

Figure 3

Exploration of the training parameters, based on $M_{\text{train}}=10000$ simulated events. (a) Estimation error $\epsilon$ as a function of the number of neurons $n_n$ in the hidden layers of the network at $n_b=50$. For $n_n=30$, we have considered a single layer or two layers with 20 and 10 neurons. (b) Estimation error $\epsilon$ for a different number of Monte Carlo repetitions $n_b$ obtained from bootstrapping at $n_n=30$. The solid line indicates the Cramér-Rao bound. (a),(b) The error bars are calculated from 35 different trainings; this takes into account the variations due to the random initialization of the neural network algorithm at the beginning of the procedure.

Figure 4

Uncertainties for parameter estimation. (a) Training set with a sampling step of 1°. (b) Training set with a sampling step of 2°; notice the logarithmic scale. (a),(b) The points refer to total numbers of events $M_{\exp }\simeq 1000$ (black solid dots), $M_{\exp }\simeq 5000$ (green squares), $M_{\exp }\simeq 10000$ (blue empty dots), and $M_{\exp }\simeq 40000$ (red diamonds); the curves are the Cramér-Rao bounds for the same number of events.