A Simple GPU-Accelerated Solver for the Schr\"odinger Operator with Applications to Ground States and Hamiltonian Simulation

TL;DR

This paper introduces a GPU-accelerated solver for the Schrödinger operator that efficiently handles separable and non-separable potentials, enabling fast ground state calculations and Hamiltonian simulations in high dimensions.

Contribution

The authors extend a tensor-product solver to the Schrödinger operator, providing efficient inversion, exponentiation, and preconditioning techniques on GPUs with theoretical guarantees.

Findings

Solver performs in less than one second for 10^9 degrees of freedom in 3D.

Preconditioned conjugate gradient method has mesh- and domain-independent iteration counts.

Applied to ground state computation and Hamiltonian simulation in high dimensions.

Abstract

We extend the tensor-product direct solver from the Laplacian to the Schr\"odinger operator $-\Delta + V$. When the potential $V_1$ is separable, the operator $-\Delta + V_1$ is inverted or exponentiated at cost $O(N^{1+1/d})$ in $d$ dimensions via per-axis eigendecomposition. On a single NVIDIA A100 GPU, this costs less than one second for $10^9$ degrees of freedom in 3D. For non-separable potentials $V = V_1 + V_2$, the same solver provides a preconditioner $(-\Delta + V_1)^{-1}$ for the preconditioned conjugate gradient (PCG) method and a propagator for operator-splitting time integrators. For bounded $V_2$, we prove that the preconditioned operator has a bounded condition number and a clustered spectrum with at most finitely many outlier eigenvalues, independently of the mesh size, and also independently of the domain size when $V_1$ is a confining potential. This explains the mesh- and domain-independent PCG iteration counts observed in practice. We apply this method to ground state computation via inverse iteration for linear problems and via the $a_u$ gradient flow for Gross--Pitaevskii energy in 3D, and also Hamiltonian simulation via the approximated qHOP and Magnus-2 splitting methods from 3D to 9D on a single NVIDIA GH200 GPU.