A Simple GPU-Accelerated Solver for the Schrödinger Operator with Applications to Ground States and Hamiltonian Simulation
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2026
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| author | Liu, Xinyu Zhang, Xiangxiong |
| author_facet | Liu, Xinyu Zhang, Xiangxiong |
| contents | We extend the tensor-product direct solver from the Laplacian to the Schrödinger operator $-Δ+ V$. When the potential $V_1$ is separable, the operator $-Δ+ V_1$ 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 $(-Δ+ V_1)^{-1}$ 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. |
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arxiv_https___arxiv_org_abs_2605_20491 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Simple GPU-Accelerated Solver for the Schrödinger Operator with Applications to Ground States and Hamiltonian Simulation Liu, Xinyu Zhang, Xiangxiong Numerical Analysis Computational Physics Quantum Physics 65M70, 65N35, 65F15, 81-08, 35Q55, 81Q05 We extend the tensor-product direct solver from the Laplacian to the Schrödinger operator $-Δ+ V$. When the potential $V_1$ is separable, the operator $-Δ+ V_1$ 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 $(-Δ+ V_1)^{-1}$ 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. |
| title | A Simple GPU-Accelerated Solver for the Schrödinger Operator with Applications to Ground States and Hamiltonian Simulation |
| topic | Numerical Analysis Computational Physics Quantum Physics 65M70, 65N35, 65F15, 81-08, 35Q55, 81Q05 |
| url | https://arxiv.org/abs/2605.20491 |