  Quantum simulation of many-body systems, particularly using ultracold atoms
and trapped ions, presents a unique form of quantum control -- it is a direct
implementation of a multi-qubit gate generated by the Hamiltonian. As a
consequence, it also faces a unique challenge in terms of benchmarking, because
the well-established gate benchmarking techniques are unsuitable for this form
of quantum control. Here we show that the symmetries of the target many-body
Hamiltonian can be used to benchmark and characterize experimental errors in
the quantum simulation. We consider two forms of errors: (i) unitary errors
arising out of systematic errors in the applied Hamiltonian and (ii) canonical
non-Markovian errors arising out of random shot-to-shot fluctuations in the
applied Hamiltonian. We show that the dynamics of the expectation value of the
target Hamiltonian itself, which is ideally constant in time, can be used to
characterize these errors. In the presence of errors, the expectation value of
the target Hamiltonian shows a characteristic thermalization dynamics, when it
satisfies the operator thermalization hypothesis (OTH). That is, an oscillation
in the short time followed by relaxation to a steady-state value in the long
time limit. We show that while the steady-state value can be used to
characterize the coherent errors, the amplitude of the oscillations can be used
to estimate the non-Markovian errors. We develop scalable experimental
protocols to characterize these errors.
