Experimental property-reconstruction in a photonic quantum extreme learning machine

Abstract

Recent developments have led to the possibility of embedding machine learning tools into experimental platforms to address key problems, including the characterization of the properties of quantum states. Leveraging on this, we implement a quantum extreme learning machine in a photonic platform to achieve resource-efficient and accurate characterization of the polarization state of a photon. The underlying reservoir dynamics through which such input state evolves is implemented using the coined quantum walk of high-dimensional photonic orbital angular momentum, and performing projective measurements over a fixed basis. We demonstrate how the reconstruction of an unknown polarization state does not need a careful characterization of the measurement apparatus and is robust to experimental imperfections, thus representing a promising route for resource-economic state characterisation.

Figures (6)

• Figure 1: Experimental QELM. (a) Schematic overview of the experimental QELM. Initial quantum states $|\psi_1 \rangle,|\psi_2 \rangle, \cdots, |\psi_n \rangle$ encoded in the polarization degree of freedom of single photons evolve through a reservoir dynamic, in which the information encoded in the initial two-dimensional space is transferred into the larger Hilbert space of the OAM. By performing only projective measurements on the OAM computational basis, the QELM is trained to reconstruct a set of target values $y_1,y_2, \cdots, y_n$. (b) Experimental implementation. Single photons, generated at 808 nm via spontaneous parametric down-conversion, are sent through the state-preparation stage ( input layer) made by a Polarizing-Beam Splitter (PBS), a Half-Wave Plate (HWP) and a Quarter-Wave Plate (QWP) to encode the initial state in the polarization degree of freedom. Subsequently, the input states evolve through the hidden layer following the quantum walk dynamics implemented by HWPs, QWPs, and Q-Plates (QPs). After projecting onto the polarization state $|\psi_{\rm pol}\rangle$ with a sequence of HWP, QWP, and PBS, projective measurements in the OAM computational basis, $\mathcal{B}=\{\ket{n}\}$ with $n=\{-2,..,2\}$, are performed through a Spatial Light Modulator (SLM) followed by the coupling into a single-mode fiber. From the counts measured by an Avalanche Photodiode (APD), the output layer of the QELM is trained to retrieve the expectation values of the observables $\{\sigma_x, \sigma_y,\sigma_z \}$.
• Figure 2: Experimental results. Estimation MSE obtained by training and testing the QELM with experimental data. The target is estimating the expectation values of the Pauli matrices, $\sigma_x,\sigma_y,\sigma_z$ on the input polarization state. We study the MSE as a function of the number of training states $N_{\rm train}$, at fixed statistics $N$. To test the protocol, we generated $300$ random input states, and tested the estimation when the first $1\le N_{\rm train} \le 150$ are used to train the QELM. The set of $300$ states remains unchanged throughout all experiments. The last $150$ of these $300$ states are always used for testing, to compute the MSE. All the points in the saturated regions of these figures decrease as $1/N$ when increasing the statistics with which each training and test state is measured. (a) Average of the MSE estimated for all three target observables: $\{\sigma_x,\sigma_y,\sigma_z\}$. We show the results for both optimized and random setups. (b) MSE for each individual target observable for the optimized setup. (c) MSE for each individual target observable for the random setup. The reported results are obtained with average experimental statistics of $\sim 3000$ counts.
• Figure S1: Smoothness of reconstruction variance We show the average estimation variance with respect to small variations of individual parameters characterizing the experimental apparatus. Each parameter is perturbed independently, leaving all the other parameters fixed at their optimal value. This showcases how different parameters affect the overall estimation accuracies in possibly different ways.
• Figure S2: MSE vs statistics. Reconstruction MSE for the observables $\sigma_x,\sigma_y,\sigma_z$ averaged over the training states, measuring states with the optimal experimental configuration. This data corresponds to the optimal measurement setup described in \ref{['sec:optimal_measurement']}. (a) MSE for each of the three target observables, for each batch of collected data. Each batch contains $300$ states, half of which are used to train the QELM, and the remaining half is used to compute the reported MSE. Each batch contains the experimental counts obtained measuring the same set of $300$ states, and they therefore only differ due to statistical fluctuations and thermal fluctuations potentially affecting the alignment of the apparatus. Training and testing are performed independently in each batch. (b) MSE for each of the three target observables, where we merge the data in the $12$ batches to study how the MSE changes with the amount of collected statistics. We cumulatively merge the statistical data for each of the $300$ states used in all $12$ batches, thus simulating an experiment where each of the states has been measured with longer and longer acquisition times. The training and testing is then performed on the resulting data.
• Figure S3: Estimated vs true expectation values. Direct comparison between true and estimated expectation values for the three Pauli observables. For each input state $\rho$, we show in red the point $(\Tr(\sigma_x \rho),\Tr(\sigma_y \rho),\Tr(\sigma_z \rho))\in\mathbb{R}^3$, while the connected blue point at the end of each arrow shows the reconstruction with the trained QELM. The data shown here corresponds to the first $100$ test states, using all training states to train the QELM, with the experimental data obtained with the optimal configuration discussed in \ref{['sec:optimal_measurement']}, using all the available statistics obtained with the $12$ batches.