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：神经网络方法逼近量子哈密顿量基态需要求解高度非线性优化问题。我们提出一种统计学习方法，通过核方法使优化过程变得简单。该方案是幂法的近似实现，利用监督学习学习幂迭代的下一步。我们证明，在监督学习高效的假设下，任意有能隙的量子哈密顿量的基态性质可以在多项式资源内获得。通过核岭回归，我们提供了数值证据，表明该学习假设在我们应用于一维和二维多个原型相互作用多体量子系统基态求解方案中得到验证，展示了该方法的灵活性。