Quantum-Enhanced Data Classification with a Variational Entangled Sensor Network

Variational quantum circuits (VQCs) built upon noisy intermediate-scale quantum (NISQ) hardware, in conjunction with classical processing, constitute a promising architecture for quantum simulations, classical optimization, and machine learning. However, the required VQC depth to demonstrate a quantum advantage over classical schemes is beyond the reach of available NISQ devices. Supervised learning assisted by an entangled sensor network (SLAEN) is a distinct paradigm that harnesses VQCs trained by classical machine-learning algorithms to tailor multipartite entanglement shared by sensors for solving practically useful data-processing problems. Here, we report the first experimental demonstration of SLAEN and show an entanglement-enabled reduction in the error probability for classification of multidimensional radio-frequency signals. Our work paves a new route for quantum-enhanced data processing and its applications in the NISQ era.

Popular Summary

Future quantum computing could solve a wide range of classically intractable problems in data analysis, optimization, and simulation. A fully fledged quantum computer requires a large number of fault-tolerant quantum bits and processing units, but at present, only noisy intermediate-scale quantum (NISQ) devices are available. An emerging way forward joins NISQ hardware with classical machine learning, giving rise to promising hybrid quantum-classical algorithms capable of tackling complex problems. But NISQ hardware still falls short of what is needed to achieve a quantum advantage in practical applications. Here, we experimentally demonstrate how NISQ hardware can enjoy a quantum advantage in real-world scenarios where sensors acquire data.

We use classical machine learning to tailor the quantum entanglement shared by a network of sensors such that the error probability in classifying their acquired data becomes substantially lower than what separable classical sensors can do. This hybrid quantum-classical scheme—coined “supervised learning assisted by an entangled sensor network” (SLAEN)—thereby demonstrates a quantum advantage. In the experiment, entangled states are produced by a quantum circuit configured by a classical machine to classify global features of the signal acquired by three radio-frequency sensors. The classical machine and the quantum circuit are trained on known radio-frequency signals to obtain the optimal configuration.

While this experiment addresses the problem of entanglement-enhanced classification of radio-frequency signals, SLAEN can incorporate different genres of sensors, thereby reaping near-term quantum advantages in a variety of realms. For example, a network of entangled sensors could be trained to precisely identify molecular species for chemical sensing, to enable ultrasensitive inertial navigation, or to facilitate early disease detection and diagnostics for health care.

Figure 1

Schematics of SLAEN and classical classifier with sample data sets. (a) In SLAEN, a VQC is configured to generate an entangled probe state. In classical processing, measurement data are utilized to train a classical SVM, whose hyperplane $\mathit{w}$ is mapped to the VQC setting $\mathit{v}$ by the VQC parameter optimizer. (b) Classical classifier using only a classical SVM. (c,d) 2D data acquired by two sensors, applicable to rf-field direction classification. (e,f) 3D data acquired by three sensors, applicable to rf-field mean-amplitude classification. Circle or sphere: data point with radius representing standard deviation of estimation uncertainty. (c,e) Entangled sensors with a clear error-probability reduction over (d,f) classical separable sensors.

Figure 2

Experimental diagram. Squeezed light processed by two VBSs, each composed of two H plates, a Q plate, a PM, and a PBS, generating a three-partite entangled probe state. Each sensor comprises an EOM and a balanced homodyne measurement setup. Measurement data were acquired by an I/O device and processed on a classical computer for training and data classification. During training, classical processing controls VBSs and EOMs through the I/O device.

Figure 3

Experimental results for training SLAEN and classical classifier. Convergence of error probabilities during training for 2D data classification (a) and 3D data classification (d). Blue curves: SLAEN. Red curves: classical classifier. Horizontal dashed lines: expected error probabilities based on true hyperplanes and measurement-noise levels. Error bars: 1 standard deviation of uncertainty derived from five measurements, each with 1000 data points. Insets: VQC parameters being optimized. VBS: variable beam splitter. History of hyperplane $\{(w_1^{(n)},w_2^{(n)}),b^{(n)}\}$ during training for 2D data classification (b,c) and 3D data classification (e,f). (b,e) SLAEN; (c,f) classical classifier. Red squares: initial hyperplane parameters prior to training. Blue triangles: hyperplane parameters after training. Hexagrams: optimum hyperplane parameters. Color gradients: evolution of error probabilities during training. Green circles: samples of hyperplane parameters at every 20 (30) training steps for 2D (3D) data classification. Curves are obtained from a cubic spline data fitting. Simulated distributions of hyperplane parameters prior to training (g), at Step 100 (h), and at Step 390 (i). Blue filled circles: SLAEN hyperplanes. Red filled circles: classical-classifier hyperplanes. Hexagrams: optimum hyperplanes. Open circles: projected hyperplane parameters onto $(w_1,w_2)$ plane (grey). SLAEN’s optimized hyperplanes distribute statistically closer to optimum solutions.

Figure 4

Scaling of error probability vs margin of the data set. Blue: SLAEN. Red: classical classifier. Circle: estimated error probability based on five sets of 1000 experimental data points. Solid lines: error probabilities obtained from Monte Carlo simulations. Shades: estimated uncertainty with 1000 samples. SLAEN shows an error-probability scaling advantage over the classical classifier.

Figure 5

Detailed experimental diagram.

Figure 6

Calibration of the variable beam splitter. (a) Light power delivered to Sensor 1 from VBS 1 (blue crosses) with respect to the input voltage to the high-voltage amplifier, which is proportional to $E_{S_1}$ in Eq. (b1). Red line: a sinusoidal fit to $v_1$ in Eq. (b1), up to a scaling factor. (b) Calibration of quadrature displacement introduced by Sensor 1 vs input voltage to the high-voltage amplifier. Blue crosses: homodyne measurement data at Sensor 1. Red solid line: a fit to $\sqrt{v_1}$ in Eq. (b1), up to a scaling factor. Error bars represent the standard deviations of the measurement results, which are determined by the shot-noise level. SNU: shot-noise unit.

Figure 7

Calibration of rf-photonic transduction. (a) Calibration of the linearity between the quadrature displacement and the phase of the rf signal proportional to the phase-modulation voltage applied on the function generator. (b) Calibration of the linearity between the quadrature displacement and the amplitude of the rf signal proportional to the amplitude-modulation voltage applied on the function generator. Blue crosses: experimental data acquired from homodyne measurements. Red curve: a linear fit. Error bars represent the standard deviations of the measurement results, which are determined by the shot-noise level. SNU: shot-noise unit.

Figure 8

Flowchart of the training process for SLAEN.

Figure 9

Experimental optimization of hyperplane parameters for a general data-classification problem. Trajectories of hyperplane parameters for SLAEN (a) and for the classical classifier (b). Red squares: initial hyperplane parameters prior to training. Blue triangles: hyperplane parameters after training. Magenta hexagrams: optimum hyperplane parameters. Color gradients: evolution of the error probabilities during training. Green circles: samples of hyperplane parameters at every 30 training steps. Curves are obtained by a cubic spline data fitting. (c) Error probabilities derived under the hyperplane parameters during training. Blue curve: error probabilities for SLAEN. Red curve: error probabilities for classical classifier.

Figure 10

Comparison between experimental data and simulations in training for 2D data classification. (a,d) Convergence of the error probabilities during training. Red curve: classical classifier. Blue curve: SLAEN. Horizontal dashed lines: expected error probabilities based on true hyperplanes and measurement-noise levels. Error bars represent 1 standard deviation of the uncertainty derived from five measurements or simulations, each with 1000 data points. (b,e) History of hyperplane parameters for SLAEN during training. (d,f) History of hyperplane parameters for classical classifier. Red squares: initial hyperplane parameters prior to training. Blue triangles: hyperplane parameters after training. Magenta hexagrams: true hyperplane parameters, representing the optimum. Color gradients: evolution of the error probabilities during training. Green circles: samples of hyperplane parameters at every 20 training steps. Curves obtained by cubic spline data fitting. Simulated distribution of hyperplane parameters prior to training (g), at Step 50 (h), and at Step 200 (i). Blue filled circles: SLAEN hyperplanes. Red filled circles: classical-classifier hyperplanes. Hexagrams: optimum hyperplanes. Open circles: projected hyperplane parameters onto the $(w_1,w_2)$ face drawn in grey. SLAEN’s optimized hyperplanes are distributed statistically closer to the optimum solutions.

Figure 11

Distance between the true hyperplane and the optimized hyperplanes at different steps under training for 2D data classification. Solid lines: distance averaged over 200 trajectories for the hyperplane parameters, each with a randomly generated initial hyperplane. Shaded area: standard deviation of the distances of 200 trajectories. Inset: zoom-in view of the distances between training Step 150 and Step 200. Blue: SLAEN. Red: classical classifier.

Figure 12

Comparison between experimental data and simulation results of training for 3D data classification. (a,d) Convergence of the error probabilities during training. Red curve: classical classifier. Blue curve: SLAEN. Horizontal dashed lines: expected error probabilities based on true hyperplanes and measurement-noise levels. Error bars represent 1 standard deviation of the uncertainty derived from five measurements or simulations, each with 1000 data points. (b,e) History of hyperplane parameters for SLAEN during training. (d,f) History of hyperplane parameters for classical classifier during training. Red squares: initial hyperplane parameters prior to training. Blue triangles: hyperplane parameters after training. Magenta hexagrams: optimum hyperplane parameters. Color gradients: evolution of the error probabilities during training. Green circles: samples of hyperplane parameters at every 30 training steps. Curves are obtained by cubic spline data fitting.

Figure 13

Distance between the true hyperplane and the optimized hyperplanes at different steps under training for 3D data classification. Solid lines: distance averaged over 200 trajectories for the hyperplane parameters, each with a randomly generated initial hyperplane. Shaded area: standard deviation of the distances of 200 trajectories. Inset: zoomed-in view of the distances between training Step 350 and 390. Blue: SLAEN. Red: classical classifier.

Figure 14

Convergence of error probability. Blue curve: simulation of training process based on separable squeezed states. Error bars represent 1 standard deviation of the uncertainty derived from 60 000 simulated data points. Red curve: SLAEN experiment. Error bars represent 1 standard deviation of the uncertainty derived from 10 measurements. Horizontal dashed lines: expected error probabilities based on true hyperplanes and measurement-noise levels.

Figure 15

Histograms of homodyne data. Red bins: separable squeezed states. Blue bins: entangled state. Histograms are normalized to probability mass functions and fitted with Gaussian probability density functions. Red curve: theory fit for separable squeezed states. Blue curve: theory fit for entangled state. Black curve: standard quantum limit.