Abstract
This work investigates the dynamical properties of classical and quantum random walks on mean-field small-world (MFSW) networks in the continuous time version. The adopted formalism profits from the large number of exact mathematical properties of their adjacency and Laplacian matrices. Exact expressions for both transition probabilities in terms of Bessel functions are derived. Results are compared to numerical results obtained by working directly the Hamiltonian of the model. For the classical evolution, any infinitesimal amount of disorder causes an exponential decay to the asymptotic equilibrium state, in contrast to the polynomial behavior for the homogeneous case. The typical quantum oscillatory evolution has been characterized by local maxima. It indicates polynomial decay to equilibrium for any degree of disorder. The main finding of the work is the identification of a faster classical spreading as compared to the quantum counterpart. It stays in opposition to the well known diffusive and ballistic for, respectively, the classical and quantum spreading in the linear chain.
Highlights
This work investigates the dynamical properties of classical and quantum random walks on meanfield small-world (MFSW) networks in the continuous time version
In this work we investigate the dynamical properties of CTQW and continuous time transport by random walkers (CTRW) on the so-called mean-field small-world (MFSW) networks[26]
The matrix AMF represents a tight-binding Hamiltonian for a quantum particle system, for which the CTQW dynamics is described by the Laplacian matrix
Summary
This work investigates the dynamical properties of classical and quantum random walks on meanfield small-world (MFSW) networks in the continuous time version. For CTQW’s, the unitary time evolution operator of probability transition between two quantum states is an exponential function of the Laplacian matrix representing the substrate. This approach is similar to the one used to describe continuous time transport by random walkers (CTRW) in classical non equilibrium statistical physics[9]. The resulting mathematical formulation, akin to that of tight-binding Hamiltonian models, reflects the similarity between time-evolution operators in statistical and in quantum mechanics. Within this analogy, CTQW stands as a linear problem, benefitting from many CTRW general results, as eigenvalue and eigenvector properties. The reliability of our approach is illustrated through the comparison with numerical results obtained by working directly the Hamiltonian of the model
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