  In this paper, we investigate the simulation of continuous-time quantum walks
on specific classes of graphs, for which it is possible to fast-forward the
time-evolution operator to achieve constant-time simulation complexity and to
perform the simulation exactly, while maintaining $\mbox{poly}(\mbox{log}(n))$
efficiency. In particular, we discuss two classes of composite graphs,
commuting graphs and Cartesian product of graphs, that contain classes of
graphs which can be simulated in this fashion. This allows us to identify new
families of graphs that we can efficiently simulate in a quantum circuit
framework, providing practical and explicit means to explore quantum-walk based
algorithms in laboratories.
