We borrow the idea that "stochastic control theory can provide a very simple model simulating quantum-mechanical behaviour" from the work of Guerra Morato [1]. In the discrete setting of a continuous-time quantum walk on a graph [2], we implement, in the numerical simulations, the prescriptions of Guerra and Marra [3]. In doing so, we endow quantum walks on a graph with the notion of trajectory, in the sense of sample path of an associated Markov process. We explore, at a frankly heuristic level, the intuition that can be gained about the behaviour of a quantum walk from the examination of its Markovian counterpart. In particular the notions of sojourn times and first passage times are introduced for a quantum walk.
[1] F. Guerra, L. Morato "Quantization of dynamical systems and stochastic control theory" Phys. Rev. D 27, 1774-1786 (1983)
[2] A. Childs, E. Farhi, S. Gutmann "An example of the difference between quantum and classical random walks" Quantum Information processing 1, 35-43 (2002)
[3] F. Guerra, R. Marra "Discrete variational principles and quantum mechanics" Phys. Rev. D 29, 1647-1665 (1984)
[1] F. Guerra, L. Morato "Quantization of dynamical systems and stochastic control theory" Phys. Rev. D 27, 1774-1786 (1983)
[2] A. Childs, E. Farhi, S. Gutmann "An example of the difference between quantum and classical random walks" Quantum Information processing 1, 35-43 (2002)
[3] F. Guerra, R. Marra "Discrete variational principles and quantum mechanics" Phys. Rev. D 29, 1647-1665 (1984)