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Cosmological simulations describing the evolution of density perturbations ofa self-gravitating collisionless Dark Matter (DM) fluid in an expandingbackground, provide a powerful tool to follow the formation of cosmicstructures over wide dynamic ranges. The most widely adopted approach, based onthe N-body discretization of the collisionless Vlasov-Poisson (VP) equations,is hampered by an unfavorable scaling when simulating the wide range of scalesneeded to cover at the same time the formation of single galaxies and of thelargest cosmic structures. The dynamics described by the VP equations islimited by the rapid increase of the number of resolution elements which isrequired to simulate an ever growing range of scales. Recent studies showed aninteresting mapping of the 6-dimensional+1 (6D+1) VP problem into a moreamenable 3D+1 non-linear Schr\"odinger-Poisson (SP) problem for simulating theevolution of DM perturbations. This opens up the possibility of improving thescaling of time propagation simulations using quantum computing. In this paper,we introduce a quantum algorithm for simulating the (SP) equation by adapting avariational real-time evolution approach to a self-consistent, non-linear,problem. To achieve this, we designed a novel set of quantum circuits thatestablish connections between the solution of the original Poisson equation andthe solution of the corresponding time-dependent Schr\"odinger equation. Wealso analyzed how nonlinearity impacts the variance of observables.Furthermore, we explored how the spatial resolution behaves as the SP dynamicsapproaches the classical limit and discovered an empirical logarithmicrelationship between the required number of qubits and the scale of the SPequation. This entire approach holds the potential to serve as an efficientalternative for solving the Vlasov-Poisson (VP) equation by means of classicalalgorithms.

From Vlasov-Poisson to Schr\"odinger-Poisson: dark matter simulation with a quantum variational time evolution algorithm