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Quantum simulators have made a remarkable progress towards exploring thedynamics of many-body systems, many of which offer a formidable challenge toboth theoretical and numerical methods. While state-of-the-art quantumsimulators are in principle able to simulate quantum dynamics well outside thedomain of classical computers, they are noisy and limited in the variability ofthe initial state of the dynamics and the observables that can be measured.Despite these limitations, here we show that such a quantum simulator can beused to in-effect solve for the dynamics of a many-body system. We develop anefficient numerical technique that facilitates classical simulations in regimesnot accessible to exact calculations or other established numerical techniques.The method is based on approximations that are well suited to describelocalized one-dimensional Fermi-Hubbard systems. Since this new method does nothave an error estimate and the approximations do not hold in general, we use aneutral-atom Fermi-Hubbard quantum simulator with $L_{\text{exp}}\simeq290$lattice sites to benchmark its performance in terms of accuracy and convergencefor evolution times up to $700$ tunnelling times. We then use theseapproximations in order to derive a simple prediction of the behaviour ofinteracting Bloch oscillations for spin-imbalanced Fermi-Hubbard systems, whichwe show to be in quantitative agreement with experimental results. Finally, wedemonstrate that the convergence of our method is the slowest when theentanglement depth developed in the many-body system we consider is neither toosmall nor too large. This represents a promising regime for near-termapplications of quantum simulators.

Benchmarking a novel efficient numerical method for localized 1D Fermi-Hubbard systems on a quantum simulator