  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.
