Quantum machine learning of graph-structured data

Graph structures are ubiquitous throughout the natural sciences. Here we develop an approach that exploits the quantum source's graph structure to improve learning via an arbitrary quantum neural network (QNN) ansatz. In particular, we devise and optimize a self-supervised objective to capture the information-theoretic closeness of the quantum states in the training of a QNN. Numerical simulations show that our approach improves the learning efficiency and the generalization behavior of the base QNN. On a practical note, scalable quantum implementations of the learning procedure described in this paper are likely feasible on the next generation of quantum computing devices.

Figure 1

Graph and QNN. Illustration of the semisupervised learning of a graph-structured quantum source (supervised nodes are shaded) via a quantum feedforward neural network (QNN). Neighboring vertices in the graph are associated with similar output states ${\rho }_x^{\text{sv}}$ or ${\rho }_x^{\text{test}}$. The QNN consists of input, an output, and $L$ hidden layers. The order of application of the perceptron unitaries is indicated with over- or undercrossings.

Figure 2

Connected clusters. The output states of this training data set comprise a graph with two clusters. The states are chosen in a way such that $v_1,...,v_4$ form one cluster, $v_8$ connects the two clusters, and the three remaining vertices form the second cluster. The coefficients are only recorded to three decimal places. Note that only $S$ of these states are used for training. In the figure $S=3$ example supervised vertices are shaded. The input states are generally taken to be unstructured. In our case the states $|{\varphi }_i^{\text{in}}\rangle$ are random three-qubit states built via a normal (Gaussian) distribution.

Figure 3

Numerical results: connected clusters. We trained a network with three supervised training pairs from Fig. 2 and the graph structure presented there. (a) Testing loss during 1000 training epochs ($\epsilon =0.01$) optimizing $\gamma =0$ supervised (blue circles) and $\gamma =-0.5$ supervised plus graph-based unsupervised objectives (red circles). (b) Testing loss after 1000 training epochs with the same data and parameters as in (a) but averaged over 30 random initializations and randomly chosen train-test splits plotted for different supervised pairs $S$. The error bars represent one standard error of the mean.

Figure 4

Line. The output states of this training data set form a line graph. The states were chosen to be, according to the fidelity, evenly spaced along a line between the states $|0\rangle$ and $|1\rangle$ associated with the endpoints. Again, note that only $S$ of these states are used for training. (Here $S=3$ example vertices are shaded.) The input states are again unstructured: $|{\varphi }_i^{\text{in}}\rangle$ are random three-qubit states built from a normal (Gaussian) distribution.

Figure 5

Numerical results: line. We trained a network with three supervised training pairs from Fig. 4 and the graph structure presented there. (a) Testing loss during 1000 epoches of training ($\epsilon =0.01$) with $\gamma =0$ semisupervised (blue $•$) and $\gamma =-1$ semisupervised plus graph information (red $•$). (b) Testing loss after 1000 epochs of training with the same data and parameters as in (a) but averaged over 30 shots plotted for different $S$ (supervised pairs). The error bars represent one standard error of the mean.

Figure 6

Numerical results: synthetic graph with nonrandom input states. The plot describes the generalization behavior of a dissipative QNN (2000 training epochs $\epsilon =0.01$) trained without (blue $•$) and with (red $•$) using the graph structure of a graph with 32 vertices produced by a classical deep walk. Each data point demonstrates an average over ten independent training attempts. The error bars represent one standard error of the mean.