  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.
