  Closed quantum systems follow a unitary time evolution that can be simulated
on quantum computers. By incorporating non-unitary effects via, e.g.,
measurements on ancilla qubits, these algorithms can be extended to open-system
dynamics, such as Markovian processes described by the Lindblad master
equation. In this paper, we analyze the convergence criteria and the
convergence speed of a Markovian, purely dissipative quantum random walk on an
unstructured search space. We analytically derive the mixing time for
continuous- and discrete-time processes for different coupling regimes that
control the random walk. Then, we map the results onto an actual implementation
to correctly estimate the potential and show that it is no more efficient than
classical search. Despite this, we highlight the connections between the
classical random walks and open quantum random walks and demonstrate how to
recover a quadratic speedup in the unitary limit. Moreover, the study of the
Grover problem reveals how the hyperparameters affect the cost of
implementation. This helps us refine the complexity analysis of the
investigated algorithm to simulate the Lindblad equation.
