Quantum Simulation on Riemannian Manifolds

2026-10-02

AI Summary

This study addresses the absence of efficient quantum simulation algorithms for the Schrödinger equation on Riemannian manifolds by proposing a novel framework that integrates global spectral methods with local coherent simulation. Methodologically, this work introduces a multi-chart coherent simulation architecture that leverages the Laplace–Beltrami operator and spectral truncation techniques to realize quantum evolution on arbitrary compact manifolds. Furthermore, it extends the quantum Hamiltonian descent method to the optimization of geodesically convex functions. Experimental evaluations conducted on spaces such as spheres and simplices validate the algorithmic efficiency of the proposed approach. Its successful application to physical simulation and optimization scenarios establishes a new paradigm for quantum computing on 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.

Problem

• Quantum Simulation
• Riemannian Manifolds
• Schrödinger Equation
• Laplace-Beltrami Operator

Innovation

• Quantum Simulation
• Riemannian Manifolds
• Laplace-Beltrami Operator
• Spectral Methods
• Quantum Hamiltonian Descent