In papers [1,2], the amplitudes of continuous-time quantum walk on graphs possessing quantum decomposition (QD graphs) have been calculated by a new method based on spectral distribution associated to their adjacency matrix. Here in this paper, it is shown that the continuous-time quantum walk on any arbitrary graph can be investigated by spectral distribution method, simply by using Krylov subspace-Lanczos algorithm to generate orthonormal bases of Hilbert space of quantum walk isomorphic to orthogonal polynomials. Also new type of graphs possessing generalized quantum decomposition have been introduced, where this is achieved simply by relaxing some of the constrains imposed on QD graphs and it is shown that both in QD and GQD graphs, the unit vectors of strata are identical with the orthonormal basis produced by Lanczos algorithm.Moreover, it is shown that probability amplitude of observing walk at a given vertex is proportional to its coefficient in the corresponding unit vector of its stratum, and it can be written in terms of the amplitude of its stratum. Finally the capability of Lanczos-based algorithm for evaluation of walk on arbitrary graphs ( GQD or non-QD types), has been tested by calculating the probability amplitudes of quantum walk on some interesting finite (infinite) graph of GQD type and finite (infinite) path graph of non-GQD type, where the asymptotic behavior of the probability amplitudes at infinite limit of number of vertices, are in agreement with those of central limit theorem of Ref. [3].

Inspired by the recent developments in the fields of quantum distributed computing, quantum systems are analyzed as networks of quantum nodes to reduce the complexity of the analysis. This gives rise to the distributed quantum consensus algorithms. Focus of this paper is on optimizing the convergence rate of the continuous time quantum consensus algorithm over a quantum network with N qudits. It is shown that the optimal convergence rate is independent of the value of d in qudits. First by classifying the induced graphs as the Schreier graphs, they are categorized in terms of the partitions of integer N . Then establishing the intertwining relation between one level dominant partitions in the Hasse Diagram of integer N , it is proved that the spectrum of the induced graph corresponding to the dominant partition is included in that of the less dominant partition. Based on this result, the proof of the Aldous' conjecture is extended to all possible induced graphs and the original optimization problem is reduced to optimizing spectral gap of the smallest induced graph. By providing the analytical solution to semidefinite programming formulation of the obtained problem, closed-form expressions for the optimal results are provided for a wide range of topologies.

An algorithm for the calculation of the resistance between two arbitrary nodes in an arbitrary distance-regular resistor network is provided, where the calculation is based on stratification introduced in [1] and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. It is shown that the resistances between a node α and all nodes β belonging to the same stratum with respect to the α (R αβ (i) , β belonging to the i-th stratum with respect to the α) are the same. Also, the analytical formulas for two-point resistances R αβ (i) , i = 1, 2, 3 are given in terms of the the size of the network and corresponding intersection numbers. In particular, the two-point resistances in a strongly regular network are given in terms of the its parameters (v, κ, λ, µ). Moreover, the lower and upper bounds for two-point resistances in strongly regular networks are discussed.

Motivated by the recent advances in the field of quantum computing, quantum systems are modelled and analyzed as networks of decentralized quantum nodes which employ distributed quantum consensus algorithms for coordination. In the literature, both continuous and discrete time models have been proposed for analyzing these algorithms. This paper aims at optimizing the convergence rate of the discrete time quantum consensus algorithm

Finding optimal weights for the problem of Fastest Distributed Consensus on networks with different topologies has been an active area of research for a number of years. Here in this work we present an analytical solution for the problem of Fastest Distributed Consensus for a network formed from fusion of two different symmetric star networks or in other words a network consists of two different symmetric star networks which share the same central node. The solution procedure consists of stratification of associated connectivity graph of network and Semidefinite Programming (SDP), particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Some numerical simulations are carried out to investigate the trade-off between the parameters of two fused star networks, namely the length and number of branches.