Predicting properties of quantum systems by regression on a quantum computer

Quantum computers can be considered as a natural means for performing machine learning tasks for labeled data which are inherently quantum. Many quantum machine learning techniques have been developed for solving classification problems, such as distinguishing between phases of matter or quantum processes. Similarly, one can consider a more general problem of regression, when the task is to predict continuous labels quantifying some property of quantum states, such as purity or entanglement. In this work, we propose a data-agnostic method for predicting such properties. The method is based on the notion of parametrized quantum circuits, and it seeks to find an observable the expectation of which gives the estimation of the property of interest with presumably low variance. We numerically test our approach in learning to predict (i) the parameter of a parametrized channel given its output state, (ii) entanglement of two-qubit states, and (iii) the parameter of a parametrized Hamiltonian given its ground state. The results show that the proposed method is able to find observables such that they provide highly accurate predictions of the considered properties, and in some cases even saturate the Cramer-Rao bound, which characterizes the prediction error.