Experimental Machine Learning of Quantum States

Quantum information technologies provide promising applications in communication and computation, while machine learning has become a powerful technique for extracting meaningful structures in “big data.” A crossover between quantum information and machine learning represents a new interdisciplinary area stimulating progress in both fields. Traditionally, a quantum state is characterized by quantum-state tomography, which is a resource-consuming process when scaled up. Here we experimentally demonstrate a machine-learning approach to construct a quantum-state classifier for identifying the separability of quantum states. We show that it is possible to experimentally train an artificial neural network to efficiently learn and classify quantum states, without the need of obtaining the full information of the states. We also show how adding a hidden layer of neurons to the neural network can significantly boost the performance of the state classifier. These results shed new light on how classification of quantum states can be achieved with limited resources, and represent a step towards machine-learning-based applications in quantum information processing.

Figure 1

ANN-based quantum state classifier. (a) Linear ANN optimizes new weight coefficients $\{{\omega }_i\}$ with 4 observables $\{⟨a_0{b}_0⟩,⟨a_0{b}_0^{\prime }⟩,⟨a_0^{\prime }{b}_0⟩,⟨a_0^{\prime }{b}_0^{\prime }⟩\}$ as input features. (b) ANN with hidden layers. To further promote the performance of the classifier, we insert an additional hidden layer, where each neuron conducts a nonlinear function on the input features.

Figure 2

Experimental setup of the quantum-state classifier. The prepared entanglement is guided to the state generator (yellow box). Either relative phases can be added between the two components or the state can be projected to four components $|H_A{H}_B⟩$, $|H_A{V}_B⟩$, $|V_A{H}_B⟩$, and $|V_A{V}_B⟩$ by the polarizers (POL) to construct the identity state. All the generated states are then analyzed by both the quantum-state tomography and the projection measurements. Theoretically predicted labels made by the PPT criteria based on the tomographic data are also added to the desired states. A quantum random number generator (QRNG) picks different components to construct mixed states as the training or the testing states. All data are sent to the agent for the training and the testing stages. DM, dichroic mirror; PBS, polarizing beam splitter.

Figure 3

Results predicted by the linear ANN. (a) Five sections correspond to five different parameters $\theta$ which set the separable-entangled bound $p$ predicted by the PPT criteria (the light gray part marks the separable section). All the data points are labeled by the linear ANN; here blue represents the entangled label while red represents the separable label. (b) Comparison of match rate between standard CHSH inequality and liner ANN classifier. The performance of optimized weight coefficients significantly surpasses the standard CHSH coefficients.

Figure 4

Performance comparison between non-linear and linear classifiers. Three classes of states $\{{\psi }_i^I\}$, $\{{\psi }_i^{II}\}$ and $\{{\psi }_i^{III}\}$ are prepared to compare the results between the non-linear and linear ANN. The gray shade gives the PPT predicted separable region. The ANN predicted labels are marked with different colors. Non-linear ANN can predict the exact margin for each kind of state while the linear ANN shows its insufficiency. The Bloch spheres at the bottom mark the tunable parameter in the three classes of states.

Figure 5

Comparison of match rates with the classifier trained by theoretical data. If the classifier is trained with theoretical data instead, in terms of the match rate, the performance (b) in predicting the experimental data is not as good as the one trained with experimental data (a). In particular, the match rate of several instances in the class $\{{\psi }_i^I\}$ cannot be improved by increasing the number of hidden neurons.