  Quantum walk is a useful model to simulate complex quantum systems and to
build quantum algorithms; in particular, to develop spatial search algorithms
on graphs, which aim to find a marked vertex as quickly as possible. Quantum
walks are interesting candidates to be implemented on quantum computers. In
this work, we describe efficient circuits that implement the evolution operator
of continuous-time quantum-walk-based search algorithms on three graph classes:
complete graphs, complete bipartite graphs, and hypercubes. For the class of
complete and complete bipartite graphs, the circuits implement the evolution
operator exactly. For the class of hypercubes, the circuit implements an
approximate evolution operator, which tends to the exact evolution operator
when the number of vertices is large. Our Qiskit simulations show that the
implementation is successful at finding the marked vertex even for
low-dimensional hypercubes.
