  We construct an efficient quantum algorithm to compute the quantum Schur-Weyl
transform for any value of the quantum parameter $q \in [0,\infty]$. Our
algorithm is a $q$-deformation of the Bacon-Chuang-Harrow algorithm, in the
sense that it has the same structure and is identically equal when $q=1$. When
$q=0$, our algorithm is the unitary realization of the Robinson-Schensted-Knuth
(or RSK) algorithm, while when $q=\infty$ it is the dual RSK algorithm together
with phase signs. Thus, we interpret a well-motivated quantum algorithm as a
generalization of a well-known classical algorithm.
