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The rapid development in hardware for quantum computing and simulation hasled to much interest in problems where these devices can exceed thecapabilities of existing classical computers and known methods. Approachingthis for problems that go beyond testing the performance of a quantum device isan important step, and quantum simulation of many-body quench dynamics is oneof the most promising candidates for early practical quantum advantage. Weanalyse the requirements for quantitatively reliable quantum simulation beyondthe capabilities of existing classical methods for analogue quantum simulatorswith neutral atoms in optical lattices and trapped ions. Considering theprimary sources of error in analogue devices and how they propagate after aquench in studies of the Hubbard or long-range transverse field Ising model, weidentify the level of error expected in quantities we extract from experiments.We conclude for models that are directly implementable that regimes ofpractical quantum advantage are attained in current experiments with analoguesimulators. We also identify the hardware requirements to reach the same levelof accuracy with future fault-tolerant digital quantum simulation. Verificationtechniques are already available to test the assumptions we make here, anddemonstrating these in experiments will be an important next step.

Propagation of errors and quantitative quantum simulation with quantum advantage