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Quantum walks in fractional diffusion processes

Path integral method has been an effective approach to solve the Fokker-Planck equation that simulates the evolution of probability distributions in diffusion processes. Yet the major challenges include the memory requirement to store the transition matrix in a fine-grained state space and a small time step required for the accurate estimation of short-time transition probabilities. Recently, continuous-time quantum walk was applied in simulating diffusion processes where simulation can be accelerated by increasing the time step size significantly. This approach is particularly helpful in studying stochastic resonance problems. In this work, a new quantum walk operator is proposed to simulate fractional diffusion processes, where Brownian motion is generalized to Levy random process. Unlike the Markov process in traditional diffusion modeling, the subdiffusive and superdiffusive random processes have strong memory effect with spatiotemporal coupling. Spatial fractional system is used in the formulation of fractional dynamics.

Author(s):

Yan Wang    
Georgia Institute of Technology
United States