  The Schur transform, which block-diagonalizes the tensor representation
$U^{\otimes n}$ of the unitary group $\mathbf{U}_d$ on $n$ qudits, is an
important primitive in quantum information and theoretical physics. We give a
generalization of its quantum circuit implementation due to Bacon, Chuang, and
Harrow (SODA 2007) to the case of mixed tensor $U^{\otimes n} \otimes
\bar{U}^{\otimes m}$, where $\bar{U}$ is the dual representation. This
representation is the symmetry of unitary-equivariant channels, which find
various applications in quantum majority vote, multiport-based teleportation,
asymmetric state cloning, black-box unitary transformations, etc. The "mixed"
Schur transform contains several natural extensions of the representation
theory used in the Schur transform, in which the main ingredient is a duality
between the mixed tensor representations and the walled Brauer algebra. Another
element is an efficient implementation of a "dual" Clebsch-Gordan transform for
$\bar{U}$. The overall circuit has complexity $\widetilde{O} ((n+m)d^4)$.
Finally, we show how the mixed Schur transform enables efficient implementation
of unitary-equivariant channels in various settings and discuss other potential
applications, including an extension of permutational quantum computing that
includes partial transposes.
