Machine learning has emerged recently as a powerful tool for predicting properties of quantum many-body systems. For many ground states of gapped Hamiltonians, generative models can learn from measurements of a single quantum state to reconstruct the state accurately enough to predict local observables. Alternatively, kernel methods can predict local observables by learning from measurements on different but related states. In this work, we combine the benefits of both approaches and propose the use of conditional generative models to simultaneously represent a family of states, by learning shared structures of different quantum states from measurements. The trained model allows us to predict arbitrary local properties of ground states, even for states not present in the training data, and without necessitating further training for new observables. We numerically validate our approach (with simulations of up to 45 qubits) for two quantum many-body problems, 2D random Heisenberg models and Rydberg atom systems.

Variational optimization of parametrized quantum circuits is an integral component for many hybrid quantum-classical algorithms, which are arguably the most promising applications of Noisy Intermediate-Scale Quantum (NISQ) computers [1]. Applications include the Variational Quantum Eigensolver (VQE) [2], Quantum Approximate Optimization Algorithm (QAOA) [3] and Quantum Neural Networks (QNNs) [4-6]. All the above are examples of stochastic optimization problems whereby one minimizes the expected value of a random cost function over a set of variational parameters, using noisy estimates of the cost and/or its gradient. In the quantum setting these estimates are obtained by repeated measurements of some Hermitian observables for a quantum state which depends on the variational parameters. A variety of optimization methods have been proposed in the variational quantum circuit literature for determining optimal variational parameters, including derivative-free (zeroth-order) methods such as Nelder-Mead, finite-differencing [7] or SPSA [8].