Learning ground states of gapped quantum Hamiltonians with Kernel Methods
This provides a more efficient alternative to neural network methods for quantum many-body systems, though it is incremental as it builds on existing kernel and power method techniques.
The authors tackled the problem of approximating ground states of gapped quantum Hamiltonians by introducing a kernel method that trivializes optimization, showing it can achieve ground state properties with polynomial resources under efficient supervised learning assumptions.
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.