Investigation of continuous-time quantum walk via spectral distribution associated with adjacency matrix

Abstract

Using the spectral distribution associated with the adjacency matrix of graphs, we introduce a new method of calculation of amplitudes of continuous-time quantum walk on some rather important graphs, such as line, cycle graph C n , complete graph K n , graph G n , finite path and some other finite and infinite graphs, where all are connected with orthogonal polynomials such as Hermite, Laguerre, Tchebichef and some other orthogonal polynomials. It is shown that using the spectral distribution, one can obtain the infinite time asymptotic behavior of amplitudes simply by using the method of stationary phase approximation(WKB approximation), where as an example, the method is applied to star, two-dimensional comb lattices, infinite Hermite and Laguerre graphs. Also by using the Gauss quadrature formula one can approximate infinite graphs with finite ones and vice versa, in order to derive large time asymptotic behavior by WKB method.Likewise, using this method, some new graphs are introduced, where their amplitude are proportional to product of amplitudes of some elementary graphs, even though the graphs themselves are not the same as Cartesian product of their elementary graphs.Finally, via calculating mean end to end distance of some infinite graphs at large enough times, it is shown that continuous time quantum walk at different infinite graphs belong to different universality classes which are also different than those of the corresponding classical ones.

Continuous-Time Quantum Walks on Trees in Quantum Probability Theory

“…2, we review the quantum probabilistic approach and give preliminaries and some examples for the walk X M . From the general theory of an interacting Fock space (see [14,17,18,19,20,21], for examples), the orthogonal polynomials {Q M satisfy the following three-term recurrence relations with a Szegö-Jacobi parameter ({ω n }, {α n }) respectively:…”

“…The walk is defined by identifying the Hamiltonian of the system with a matrix related to the adjacency matrix of the tree. Concerning continuous-time quantum walks, see [3,4,5,6,7,8,9,10,11,12,13,14,15,16] for examples.…”

Mixing and decoherence in continuous-time quantum walks on long-range interacting cycles

“…The DTQWs have been investigated on trees [8], on random environments [9], for single and entangled particles [10] and also in [1,11,12]. In the recent years, the CTQWs have been studied on n-cube [13], star graph [14,15], small-world network [16], quotient graph [17], line [18,19,20], dendrimer [21], distance regular graph [22], circulant Bunkbeds [23], odd graph [24] and on decision tree [25,26].…”

Investigation graph isomorphism problem via entanglement entropy in strongly regular graphs

“…Then we multiply the above equation from the left side in e T κ and use the equations (4-21) and (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) to prove that…”

“…So first we use the stratification techniques [15][16][17][18][19], to write the adjacency matrices of SRGs in the block form. The obtained matrix, becomes block diagonal in the stratification basis.…”