  We present quantum algorithms for routing concentration assignments on full
capacity fat-and-slim concentrators, bounded fat-and-slim concentrators, and
regular fat-and-slim concentrators. Classically, the concentration assignment
takes $O(n)$ time on all these concentrators, where $n$ is the number of
inputs. Powered by Grover's quantum search algorithm, our algorithms take
$O(\sqrt{nc}\ln{c})$ time, where $c$ is the capacity of the concentrator. Thus,
our quantum algorithms are asymptotically faster than their classical
counterparts, when $c\ln^2{c}=o(n)$.In general, $c = n^\mu,$ satisfies
$c\ln^2{c}=o(n),$ implying a time complexity of $O(n^{0.5(1+ \mu )} \ln n),$
for any $\mu, 0 < \mu < 1.$
