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 $\ell_p$ 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.

Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS)

arXiv:2610.03673 [quant-ph]

(or arXiv:2610.03673v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.03673 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.