  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 $N$.
