Learning quantum properties from short-range correlations using multi-task networks

TL;DR

Characterizing multipartite quantum states is challenging when target properties involve correlations across many particles. The paper introduces a multi-task neural-network that extracts a concise state representation from short-range measurements and uses separate decoders to predict multiple properties, including nonlocal invariants such as the string order parameter $\langle\tilde{S}\rangle$. Across cluster-Ising and bond-alternating XXZ ground states, the approach achieves high predictive accuracy and enables unsupervised phase classification via learned representations, with evidence of transfer to higher dimensions and unseen Hamiltonians, including the invariant $\mathcal{Z}_{\text{R}}$. These results suggest a practical tool for characterizing intermediate-scale quantum systems with limited local data, reducing measurement requirements while maintaining robust performance across out-of-distribution states and perturbations.

Abstract

Characterizing multipartite quantum systems is crucial for quantum computing and many-body physics. The problem, however, becomes challenging when the system size is large and the properties of interest involve correlations among a large number of particles. Here we introduce a neural network model that can predict various quantum properties of many-body quantum states with constant correlation length, using only measurement data from a small number of neighboring sites. The model is based on the technique of multi-task learning, which we show to offer several advantages over traditional single-task approaches. Through numerical experiments, we show that multi-task learning can be applied to sufficiently regular states to predict global properties, like string order parameters, from the observation of short-range correlations, and to distinguish between quantum phases that cannot be distinguished by single-task networks. Remarkably, our model appears to be able to transfer information learnt from lower dimensional quantum systems to higher dimensional ones, and to make accurate predictions for Hamiltonians that were not seen in the training.

Figures (10)

• Figure 1: Flowchart of our multi-task neural network. In the data acquisition phase (1), the experimenter performs short-range local measurements on the system of interest. The resulting data is used to produce a concise representation of the quantum state (2). The state representation is then fed into a set of prediction networks, each of which generates predictions for a given type of quantum property (3). After the state representation network and prediction networks are jointly trained, the state representations are employed in new tasks, such as unsupervised classification of quantum phases of matter, or prediction of order parameters and topological invariants (4). Once trained, the overall model can generally be applied to out-of-distribution quantum states and higher-dimensional quantum systems (5).
• Figure 2: Predicting properties of ground states of cluster-Ising model. Subfigure a compares the prediction accuracy between our multi-task model and single-task models for predicting two-point correlation functions $\mathcal{C}_{1j}^x$ and $\mathcal{C}_{1j}^z$, and entanglement entropy $\mathcal{S}_A$. Subfigures b and c show how the number of samples for each measurement and the number of measurements affect the coefficient of determination for the predictions of $\mathcal{S}_A$, $\mathcal{C}_{1j}^x$ and $\mathcal{C}_{1j}^z$, respectively. Subfigures d and e show the predictions of $S_A$ and $\mathcal{C}_{1j}^z$ for a ground state near phase transition marked by a red star in Subfigure \ref{['fig3']} d.
• Figure 3: Transfer learning to predict properties of the ground states of the cluster-Ising model. Subfigures a, b and c illustrate the 2D projection of the state representations obtained with the t-SNE algorithm, where the color of each data point indicates the true value of the string order parameter $\braket{\tilde{S}}$ of the corresponding ground state. Subfigure a corresponds to the state representations produced for jointly predicting spin correlations and entanglement entropy. Subfigures b and c correspond to the state representations produced for separately predicting entanglement entropy and spin correlations, respectively. Subfigure d shows the predictions of $\braket{\tilde{S}}$ for the ground states corresponding to a $64\times 64$ grid in parameter space, together with the true values of $\braket{\tilde{S}}$ for $100$ randomly chosen states indicated by white circles, where the dashed curves are the phase boundaries between SPT phase and the other two phases.
• Figure 4: 2D projections of state representations for those states prepared by shallow random symmetric quantum circuits. Subfigures a and b correspond quantum states in the SPT and the trivial phases prepared by one layer of random quantum gates, and Subfigures c and d correspond quantum states in the SPT and the trivial phases prepared by two layers of random quantum gates. Subfigures a and c illustrate state representations produced by our multi-task neural network. Subfigures b and d illustrate state representations produced by the neural network trained only on spin correlations.
• Figure 5: Prediction of properties of ground states of a perturbed Hamiltonian. Subfigure a illustrates the 2D projections of state representations for the ground states of the perturbed Hamiltonian, together with their true values of $\braket{\tilde{S}}$. Subfigure b illustrates the predictions of $\braket{\tilde{S}}$ using our adjusted neural network for the perturbed model. Subfigure c show the true values of string order parameters $\braket{\tilde{S}}$ for both the original model (\ref{['eq:clusterIsing']}) and the perturbed model (\ref{['eq:perturb']}).