  Quantum walks are promising tools based on classical random walks, with
plenty of applications such as many variants of optimization. Here we introduce
the semiclassical walks in discrete time, which are algorithms that combines
classical and quantum dynamics. Specifically, a semiclassical walk can be
understood as a classical walk where the transition matrix encodes the quantum
evolution. We have applied this algorithm to Szegedy's quantum walk, which can
be applied to any arbitrary weighted graph. We first have solved the problem
analytically on regular 1D cycles to show the performance of the semiclassical
walks. Next, we have simulated our algorithm in a general inhomogeneous
symmetric graph, finding that the inhomogeneity drives a symmetry breaking on
the graph. Moreover, we show that this phenomenon is useful for the problem of
ranking nodes in symmetric graphs, where the classical PageRank fails. We have
demonstrated experimentally that the semiclassical walks can be applied on real
quantum computers using the platform IBM Quantum.
