Feynman Formula for Discrete-Time Quantum Walks

Abstract We explicitly connect (discrete-time) quantum walks on $$\mathbb {Z} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> with a four-state Markov additive process via a Feynman-type formula (13). Using this representation, we derive a relation between the spectral decomposition of the Markov additive process and the limiting density of the homogeneous quantum walk. In addition, we consider a space-time rescaling of quantum walks, which leads to a system of quantum transport PDEs in continuous time and space with a phase interaction term. Our probabilistic representation for this type of PDE offers an efficient Monte Carlo computational technique.