Benchmarking an approximation hypothesis for localized 1D Fermi-Hubbard systems on a cold-atom quantum simulator

Quantum simulators have made significant progress towards simulating quantum many-body systems, which are intractable to current numerical and theoretical methods. However, state-of-the art quantum simulators are noisy and limited in the variability of the initial state of the dynamics and the observables that can be measured. Despite these limitations, here we show that such a quantum simulator can be used practically to in-effect solve for the dynamics of a many-body system. Any computation of the dynamics of quantum many-body systems is met with challenges arising from the exponentially growing dimension of the Hilbert space. A natural countermeasure is to relax the tolerance and seek approximate solutions.  A key feature of approximate methods is the error estimate, which allows us to determine when it is reliable.  However, not every approximation ansatz has a well established error estimate. We show that a neutral atom quantum simulator can be used to benchmark such theories, in the absence of an error estimate. We consider a localized 1D Fermi-Hubbard system where and develop an efficient  approximate theory. Our approximate theory does not have an error estimate and therefore, we use a quantum simulator to benchmark its performance in terms of accuracy of its outcome. Finally, we identify the nature of the many-body dynamics for which our approximate theory breaks down. This represents an opportune regime for the deployment of quantum simulators in the future.