Quantum Riemannian Hamiltonian Descent

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 at 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 work 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.