  Jointly matching multiple, non-rigidly deformed 3D shapes is a challenging,
$\mathcal{NP}$-hard problem. A perfect matching is necessarily
cycle-consistent: Following the pairwise point correspondences along several
shapes must end up at the starting vertex of the original shape. Unfortunately,
existing quantum shape-matching methods do not support multiple shapes and even
less cycle consistency. This paper addresses the open challenges and introduces
the first quantum-hybrid approach for 3D shape multi-matching; in addition, it
is also cycle-consistent. Its iterative formulation is admissible to modern
adiabatic quantum hardware and scales linearly with the total number of input
shapes. Both these characteristics are achieved by reducing the $N$-shape case
to a sequence of three-shape matchings, the derivation of which is our main
technical contribution. Thanks to quantum annealing, high-quality solutions
with low energy are retrieved for the intermediate $\mathcal{NP}$-hard
objectives. On benchmark datasets, the proposed approach significantly
outperforms extensions to multi-shape matching of a previous quantum-hybrid
two-shape matching method and is on-par with classical multi-matching methods.
