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arXiv:2603.15870

Riemannian gradient descent for Hartree-Fock theory

Evgueni Dinvay

quant-phSubmitted Mar 16, 2026• Announced Mar 18, 2026
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Abstract

We present a Riemannian optimization framework for Hartree-Fock theory formulated directly in the Sobolev space $H^1$. The orthonormality constraints are interpreted geometrically via infinite-dimensional Stiefel and Grassmann manifolds endowed with the embedded $H^1$ metric. Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived using resolvent operators, avoiding distributional formulations. The resulting algorithms include Riemannian steepest descent and a preconditioned nonlinear conjugate gradient method equipped with Armijo backtracking and Powell-type restarts. Particular attention is given to physically motivated preconditioning based on inversion of the kinetic energy operator. The framework is naturally compatible with adaptive multiwavelet discretizations, where Coulomb-type convolutions can be evaluated efficiently. Numerical experiments demonstrate robust convergence and competitive performance compared to conventional SCF-DIIS schemes. In addition, for small molecules the gradient descent method converges from random initial guesses. The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization and offers a foundation for further developments in infinite-dimensional Riemannian methods for quantum chemistry.

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We present a Riemannian optimization framework for Hartree-Fock theory formulated directly in the Sobolev space H1H^1H1. The orthonormality constraints are interpreted geometrically via infinite-dimensional Stiefel and Grassmann manifolds endowed with the embedded H1H^1H1 metric. Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived using resolvent operators, avoiding distributional formulations. The resulting algorithms include Riemannian steepest descent and a preconditioned nonlinear conjugate gradient method equipped with Armijo backtracking and Powell-type restarts. Particular attention is given to physically motivated preconditioning based on inversion of the kinetic energy operator. The framework is naturally compatible with adaptive multiwavelet discretizations, where Coulomb-type convolutions can be evaluated efficiently. Numerical experiments demonstrate robust convergence and competitive performance compared to conventional SCF-DIIS schemes. In addition, for small molecules the gradient descent method converges from random initial guesses. The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization and offers a foundation for further developments in infinite-dimensional Riemannian methods for quantum chemistry.
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We propose Quantum Riemannian Hamiltonian Descent (QRHD), a quantum algorithm for continuous optimization on Riemannian manifolds that extends Quantum Hamiltonian Descent (QHD) by incorporating geometric structure of the parameter space via a position-dependent metric in the kinetic term. We formulate QRHD in both operator and path integral formalisms and derive the corresponding quantum equations of motion, showing that quantum corrections appear in the action integral but they are suppressed at late times by the time-dependent dissipation factor. This implies that convergence near optimal points is controlled by the classical potential while quantum effects influence early-time dynamics. By analyzing the semiclassical equation, we estimate a lower bound on the convergence time and numerically demonstrate whether QRHD works as a quantum optimization algorithm in some examples. A quantum circuit implementation based on time-dependent Hamiltonian simulation is also discussed and the query complexity is estimated.