Position-dependent and cooperative quantum Parrondo walks

This work is a study on quantum computational formulations of Parrondo walks, that is, positively trending random walks formed as combinations of negatively trending random walks. We reanalyse the position-dependent walk proposed by Košík et al (2007 J. Mod. Opt. 54 2275), correcting the parameter choices in that paper to achieve the Parrondo effect. We also devise a quantum analogue of the cooperative Parrondo walk of Toral (2002 Fluct. Noise Lett. 2 L305), in which it is the interaction between multiple participants, rather than position-dependence, that allows the Parrondo effect to occur. We give a general formulation of a quantum analogue of the classical walk of Toral (2002 Fluct. Noise Lett. 2 L305), and demonstrate the Parrondo effect numerically. Lastly, we highlight a qualitative difference in asymptotic behaviour between quantum Parrondo walks and their classical counterparts. In particular, we draw attention to an intuitive but unreliable assumption, based on classical random walks, which may pose extra challenges for applications of the Parrondo effect in the quantum setting seeking to separate or classify data or particles.