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