  Simulation of the single-particle tunneling problem by means of the
Suzuki-Trotter approximation (STA) is analyzed. Considered is a particle
hopping across a chain of sites in presence of a smooth position-dependent
potential profile with several local minima that arrange a tunneling problem
between the localized states in different minima. The STA error is found to
manifest itself in three ways: i) perturbative energy shifts, ii)
nonperturbartive renormalization of the tunneling rates, and iii) perturbative
leakage of the total probability to other states. Generally, the first type of
error is the most essential, as detuning of the tunneling resonance has to be
compared with exponentially small tunneling rates. In absence of detuning (e.g.
if the resonance is protected by symmetry), STA leads to exponential
enhancement of the tunneling rates. The last type of error classifies the
overall defect in the wave function and delineates the region of sufficiently
weak distortion of the wave function due to STA. The conducted analysis
confirms the naive criteria of applicability $\max\{T,P\}\ll\delta t^{-1}$
(with $T,P$ being the typical scales of kinetic and potential terms,
respectively), while also revealing the structure of error and its behavior
with system parameters. Analysis of the case of large Trotter step is also
performed, with the main result being the reconstruction of low-energy spectrum
due to coupling between states with energy difference close to $2\pi/\delta t$.
The connection of the obtained results with rigorous upper error bounds on the
STA error is discussed, with particular emphasis on why these rigorous bounds
are not always saturated. We also point out that the proposed problem can be
directly implemented on existing quantum devices [arXiv:2012.00921]. In
particular, we give a detailed description of an experimental design that
demonstrates the described physics.
