Quantum walks in spaces with applied potentials

Discrete quantum walks are a universal model of quantum computation equivalent to the quantum circuit model and can be mapped onto quantum circuits and executed using quantum computers. Quantum walks can model and simulate many physical systems and several quantum algorithms are based on them. Discrete quantum walks have been extensively studied, but quantum walks that evolve in spaces in which potentials are applied received little or no attention. Here, we formulate the discrete quantum walk model in one and two-dimensional spaces in which potentials are applied. In this formulation the quantum walker carries a “charge” affected by the potentials and the walk evolution is driven by both constant and time-varying potentials. We reproduce the tunneling through a barrier phenomenon and study the quantum walk evolution in one and two-dimensional spaces with various potential distributions. We demonstrate that our formulation can serve as a basis for applied quantum computing by studying maze running and the motion of vehicles in urban spaces. In these spaces curbs and buildings are modeled as impenetrable potential barriers and traffic lights as time-varying potential barriers. Quantum walks in spaces with applied potentials may open the way for the development of novel quantum algorithms in which inputs are introduced as potential profiles.