  Quantum and random walks were proven to be equivalent on finite graphs by
demonstrating how to construct a time-dependent random walk sharing the exact
same evolution of vertex probability of any given discrete-time coined quantum
walk. Such equivalence stipulated a deep connection between the processes that
is far stronger than simply considering quantum walks as quantum analogues of
random walks. This article expands on the connection between quantum and random
walks by demonstrating a procedure that constructs a time-dependent quantum
walk matching the evolution of vertex probability of any given random walk in a
unitary way. It is a trivial fact that a quantum walk measured at all time
steps of its evolution degrades to a random walk. More interestingly, the
method presented describes a quantum walk that matches a random walk without
measurement operations, such that the unitary evolution of the quantum walk
captures the probability evolution of the random walk. The construction
procedure is general, covering both homogeneous and non-homogeneous random
walks. For the homogeneous random walk case, the properties of unitary
evolution imply that the quantum walk described is time-dependent since
homogeneous quantum walks do not converge for arbitrary initial conditions
