Learning ground states of gapped quantum Hamiltonians with Kernel Methods

Clemens Giuliani1,2, Filippo Vicentini1,2, Riccardo Rossi1,2,3, and Giuseppe Carleo1,2

1Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
2Center for Quantum Science and Engineering, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France

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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.

Learning the ground state of a physical system is a computationally hard task. Machine learning (ML) techniques have been proposed to solve this problem. However, the variational optimization of the wave function in these approaches is a highly nonlinear problem, which is not straightforward to solve.
In this article, we show that by using the power method as an iterative supervised learning strategy, combined with a kernel method called kernel ridge regression, this can be cast into a series of trivial-to-solve convex optimization problems. These do not require gradient-descent learning as in standard neural-network-based ML approaches.
More generally, we show that this approximate self-learning power method converges to the ground state as long as the error per step is small.

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