Quantum phase detection generalization from marginal quantum neural network models

Quantum machine learning offers a promising advantage in extracting information about quantum states, e.g., phase diagram. However, access to training labels is a major bottleneck for any supervised approach, preventing getting insights about new physics. In this Letter, using quantum convolutional neural networks, we overcome this limit by determining the phase diagram of a model where analytical solutions are lacking, by training only on marginal points of the phase diagram, where integrable models are represented. More specifically, we consider the axial next-nearest-neighbor Ising Hamiltonian, which possesses a ferromagnetic, paramagnetic, and antiphase, showing that the whole phase diagram can be reproduced.

Figure 1

Circuit architecture: VQE states (blue) are the input of the quantum convolutional neural network composed of free rotations $R$ (yellow), convolutions $C$ (light green), pooling $P$ (red), and a fully connected layer $F$ (dark green).

Figure 2

Fidelity between the ground states of the ANNNI model at $h=0.3$ and $N=12$. We observe three different clusters, corresponding to the ferromagnetic, paramagnetic, and antiphase, respectively.

Figure 3

Quantum phase classification. (a) shows the classification accuracy of the QCNN as a function of the number of training points $n$, for the Gaussian centered around the critical points ($GC$ blue), Gaussian centered around the middle of each phase ($G2$ black), and uniform sampling (red). (b) displays the phase diagram predicted by the QCNN trained on ${\mathcal{S}}_{G2}^{40}$ (red dots) where the color represents the probability mixture of being in one of the three phases and the red lines the predicted boundaries, while (c) shows the anomaly score for $N=6$ spin systems trained on the initial state $|\psi \rangle$ (red cross). (d)–(f) are similar but for $N=12$ spins. The black lines are $h_I\left(\kappa \right)$ for $\kappa <0.5$ and $h_C\left(\kappa \right)$ for $\kappa >0.5$.