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

TL;DR

This paper introduces a quantum algorithm using a variational real-time evolution approach to simulate the Schr"odinger-Poisson equations for dark matter, potentially improving the scalability of cosmological simulations compared to traditional methods.

Contribution

The authors develop a novel quantum circuit framework for solving the non-linear Schr"odinger-Poisson equations, connecting quantum solutions to classical Poisson problems and analyzing the impact of nonlinearity and resolution.

Findings

Discovered an empirical logarithmic relationship between qubits and spatial scale.

Designed quantum circuits linking Poisson and Schr"odinger solutions.

Analyzed how nonlinearity affects observable variance.

Abstract

Cosmological simulations describing the evolution of density perturbations of a self-gravitating collisionless Dark Matter (DM) fluid in an expanding background, provide a powerful tool to follow the formation of cosmic structures over wide dynamic ranges. The most widely adopted approach, based on the N-body discretization of the collisionless Vlasov-Poisson (VP) equations, is hampered by an unfavorable scaling when simulating the wide range of scales needed to cover at the same time the formation of single galaxies and of the largest cosmic structures. The dynamics described by the VP equations is limited by the rapid increase of the number of resolution elements which is required to simulate an ever growing range of scales. Recent studies showed an interesting mapping of the 6-dimensional+1 (6D+1) VP problem into a more amenable 3D+1 non-linear Schr\"odinger-Poisson (SP) problem for simulating the evolution of DM perturbations. This opens up the possibility of improving the scaling of time propagation simulations using quantum computing. In this paper, we introduce a quantum algorithm for simulating the (SP) equation by adapting a variational real-time evolution approach to a self-consistent, non-linear, problem. To achieve this, we designed a novel set of quantum circuits that establish connections between the solution of the original Poisson equation and the solution of the corresponding time-dependent Schr\"odinger equation. We also analyzed how nonlinearity impacts the variance of observables. Furthermore, we explored how the spatial resolution behaves as the SP dynamics approaches the classical limit and discovered an empirical logarithmic relationship between the required number of qubits and the scale of the SP equation. This entire approach holds the potential to serve as an efficient alternative for solving the Vlasov-Poisson (VP) equation by means of classical algorithms.