Saber Jafarizadeh

Saber Jafarizadeh

Traditionally, systems governed by linear Partial Differential Equations (PDEs) are spatially discretized to exploit their algebraic structure and reduce the computational effort for controlling them. Due to beneficial insights of the PDEs, recently, the reverse of this approach is implemented where a spatially-discrete system is approximated by a spatially-continuous one, governed by linear PDEs forming diffusion equations. In the case of distributed consensus algorithms, this approach is adapted to enhance its convergence rate to the equilibrium. In previous studies within this context, constant diffusion parameter is considered for obtaining the diffusion equations. This is equivalent to assigning a constant weight to all edges of the underlying graph in the consensus algorithm. Here, by relaxing this restricting assumption, a spatially-variable diffusion parameter is considered and by optimizing the obtained system, it is shown that significant improvements are achievable in terms of the convergence rate of the obtained spatially-continuous system. As a result of approximation, the system is divided into two sections, namely, the spatially-continuous path branches and the lattice core which connects these branches at one end. The optimized weights and diffusion parameter for each of these sections are optimal individually but considering the whole system, they are suboptimal. It is shown that the symmetric star topology is an exception and the obtained results for this topology are globally optimal. Furthermore, through variational method, the results obtained for the symmetric star topology are validated and it is shown that the variable diffusion parameter improves the robustness of the system too.

Saber Jafarizadeh

Distributed consensus algorithm over networks of quantum systems has been the focus of recent studies in the context of quantum computing and distributed control. Most of the progress in this category have been on the convergence conditions and optimizing the convergence rate of the algorithm, for quantum networks with undirected underlying topology. This paper aims to address the extension of this problem over quantum networks with directed underlying graphs. In doing so, the convergence to two different stable states namely, consensus and synchronous states have been studied. Based on the intertwining relation between the eigenvalues, it is shown that for determining the convergence rate to the consensus state, all induced graphs should be considered while for the synchronous state only the underlying graph suffices. Furthermore, it is illustrated that for the range of weights that the Aldous' conjecture holds true, the convergence rate to both states are equal. Using the Pareto region for convergence rates of the algorithm, the global and Pareto optimal points for several topologies have been provided.

Saber Jafarizadeh

Distributed gossip algorithm has been studied in literature for practical implementation of the distributed consensus algorithm as a fundamental algorithm for the purpose of in-network collaborative processing. This paper focuses on optimizing the convergence rate of the gossip algorithm for both classical and quantum networks. A novel model of the gossip algorithm with non-uniform clock distribution is proposed which can reach the optimal convergence rate of the continuous-time consensus algorithm. It is described that how the non-uniform clock distribution is achievable by modifying the rate of the Poisson process modeling the clock of the gossip algorithm. The minimization problem for optimizing the asymptotic convergence rate of the proposed gossip algorithm and its corresponding semidefinite programming formulation is addressed analytically. It is shown that the optimal results obtained for uniform clock distribution are suboptimal compared to those of the non-uniform one and for non-uniform distribution the optimal answer is not unique i.e. there is more than one set of probabilities that can achieve the optimal convergence rate. Based on the optimal continuous-time consensus algorithm and the detailed balance property, an effective method of obtaining one of these optimal answers is proposed. Regarding quantum gossip algorithm, by expanding the density matrix in terms of the generalized Gell-Mann matrices, the evolution equation of the quantum gossip algorithm is transformed to the state update equation of the classical gossip algorithm. By defining the quantum gossip operator, the original optimization problem is formulated as a convex optimization problem, which can be addressed by semidefinite programming.

Saber Jafarizadeh

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 over a quantum network with $N$ qudits. The induced graphs are categorized in terms of the partitions of integer $N$ by arranging them as the Schreier graphs. It is shown that the original optimization problem reduces to optimizing the Second Largest Eigenvalue Modulus (SLEM) of the weight matrix. Exploiting the Specht module representation of partitions of $N$, the Aldous' conjecture is generalized to all partitions (except ($N$)) in the Hasse diagram of integer $N$. Based on this result, it is shown that the spectral gap of Laplacian of all induced graphs corresponding to partitions (other than ($N$)) of $N$ are the same, while the spectral radius of the Laplacian is obtained from the feasible least dominant partition in the Hasse diagram of integer $N$. The semidefinite programming formulation of the problem is addressed analytically for $N \leq d^2 + 1$ and a wide range of topologies where closed-form expressions for the optimal results are provided. For a quantum network with complete graph topology, solution of the optimization problem based on group association schemes is provided for all values of $N$.

Saber Jafarizadeh

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.