Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using kernel ridge regression, we provide numerical evidence that the learning assumption is verified by applying our scheme to find the ground states of several prototypical interacting many-body quantum systems, both in one and two dimensions, showing the flexibility of our approach.

翻译：神经网络方法近似求解量子哈密顿量基态需要数值求解高度非线性优化问题。我们提出一种统计学习方法，通过使用核方法使优化过程变得简单。该方案实现了幂法的近似版本，其中利用监督学习来学习幂迭代的下一步骤。我们证明，在监督学习有效的假设下，任意带隙量子哈密顿量的基态性质可通过多项式资源获得。通过应用核岭回归方法寻找多个一维和二维典型相互作用多体量子系统的基态，我们提供了数值证据表明该学习假设成立，同时展示了我们方法的灵活性。