Investigation of continuous-time quantum walk by using Krylov subspace-Lanczos algorithm

摘要: In papers [Jafarizadehn and Salimi, Ann. Phys. 322, 1005 (2007) J. A: Math. Gen. 39, 13295 (2006)], the amplitudes of continuous-time quantum walk (CTQW) on graphs possessing decomposition (QD graphs) have been calculated by a new method based spectral distribution associated with their adjacency matrix. Here in this paper, it is shown that CTQW any arbitrary graph can be investigated analysis method, simply using Krylov subspace-Lanczos algorithm to generate orthonormal bases Hilbert space isomorphic orthogonal polynomials. Also type generalized (GQD) introduced, where achieved relaxing some constrains imposed QD both GQD graphs, unit vectors strata are identical basis produced Lanczos algorithm. Moreover, probability amplitude observing at given vertex proportional its coefficient corresponding vector stratum, written terms stratum. The capability Lanczos-based for evaluation (GQD or non-QD types), has tested calculating interesting finite (infinite) path non-GQD type, asymptotic behavior limit large number vertices, agreement those central theorem [Phys. Rev. E 72, 026113 (2005)]. At end, applications such as implementation search algorithms, resistance between two nodes regular networks solid state condensed matter physics, discussed, all them, algorithm, reduces smaller subspaces problem subspace maximal dimension.