  In a circuit-based quantum computer, the computing is performed via the
discrete-time evolution driven by quantum gates. Accurate simulation of
continuoustime evolution requires a large number of quantum gates and therefore
suffers from more noise. In this paper, we find that shallow quantum circuits
are sufficient to qualitatively observe some typical quantum phenomena in the
continuous-time evolution limit, such as resonant tunneling and localization
phenomena. We study the propagation of a spin excitation in Trotter circuits
with a large step size. The circuits are formed of two types of two-qubit
gates, i.e. XY gates and controlled- Rx gates, and single-qubit Rz gates. The
configuration of the Rz gates determines the distribution of the spin
excitation at the end of evolution. We demonstrate the resonant tunneling with
up to four steps and the localization phenomenon with dozens of steps in
Trotter circuits. Our results show that the circuit depth required for
qualitative observation of some significant quantum phenomena is much smaller
than that required for quantitative computation, suggesting that it is feasible
to apply qualitative observations to near-term quantum computers. We also
provide a way to use the physics laws to understand the error propagation in
quantum circuits.
