  We study the behavior of errors in the quantum simulation of spin systems
with long-range multi-body interactions resulting from the Trotter-Suzuki
decomposition of the time-evolution operator. We identify a regime where the
Floquet operator underlying the Trotter decomposition undergoes sharp changes
even for small variations in the simulation step size. This results in a time
evolution operator that is very different from the dynamics generated by the
targeted Hamiltonian, which leads to a proliferation of errors in the quantum
simulation. These regions of sharp change in the Floquet operator, referred to
as structural instability regions, appear typically at intermediate Trotter
step sizes and in the weakly-interacting regime, and are thus complementary to
recently revealed quantum chaotic regimes of the Trotterized evolution
(Sieberer et al., npj Quantum Information 5, 1 (2019)). We characterize these
structural instability regimes in $p$-spin models, transverse-field Ising
models with all-to-all $p$-body interactions, and analytically predict their
occurrence based on unitary perturbation theory. We further show that the
effective Hamiltonian associated with the Trotter decomposition of the unitary
time-evolution operator, when the Trotter-step size is chosen to be in the
structural instability region, is very different from the target Hamiltonian,
which explains the large errors that can occur in the simulation in the regions
of instability. These results have implications for the reliability of
near-term gate-based quantum simulators, and reveal an important interplay
between errors and the physical properties of the system being simulated.
