Quantum Simulation on Riemannian Manifolds
Abstract
We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and ℓp balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.
Related
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BibTeX
@misc{herman2026quantum,
title = {{Quantum Simulation on Riemannian Manifolds}},
author = {Dylan Herman and Jacob Watkins and Guneykan Ozgul and Jiayu Shen and Brandon Augustino and Junhyung Lyle Kim and Shouvanik Chakrabarti},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03673},
eprint = {2610.03673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03673},
}
RIS
TY - UNPB
TI - Quantum Simulation on Riemannian Manifolds
AU - Dylan Herman
AU - Jacob Watkins
AU - Guneykan Ozgul
AU - Jiayu Shen
AU - Brandon Augustino
AU - Junhyung Lyle Kim
AU - Shouvanik Chakrabarti
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03673
UR - https://arxiv.org/abs/2610.03673
AB - We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and ℓ_p balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.
ER -