  This paper establishes a framework for the acceleration of quantum gossip
algorithms by introducing local clique operations to networks of interconnected
qubits. Cliques are local structures in complex networks being complete
subgraphs. Based on cyclic permutations, clique gossiping leads to collective
multi-party qubit interactions. This type of algorithm can be physically
realized by a series of local environments using coherent methods. First of
all, we show that at reduced states, these cliques have the same acceleration
effects as their roles in accelerating classical gossip algorithms, which can
even make possible finite-time convergence for suitable network structures.
Next, for randomized selection of cliques where node updates enjoy a more
self-organized and scalable sequencing, we show that the rate of convergence is
precisely improved by $\mathcal{O}(k/n)$ at the reduced states, where $k$ is
the size of the cliques and $n$ is the number of qubits in the network. The
rate of convergence at the coherent states of the overall quantum network is
proven to be decided by the spectrum of a mean-square error evolution matrix.
Explicit calculation of such matrix is rather challenging, nonetheless, the
effect of cliques on the coherent states' dynamics is illustrated via numerical
examples. Interestingly, the use of larger quantum cliques does not necessarily
increase the speed of the network density aggregation, suggesting quantum
network dynamics is not entirely decided by its classical topology.
