  We present an efficient family of quantum circuits for a fundamental
primitive in quantum information theory, the Schur transform. The Schur
transform on n d-dimensional quantum systems is a transform between a standard
computational basis to a labelling related to the representation theory of the
symmetric and unitary groups. If we desire to implement the Schur transform to
an accuracy of epsilon, then our circuit construction uses a number of gates
which is polynomial in n, d and log(1/epsilon). The important insights we use
to perform this construction are the selection of the appropriate subgroup
adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction
renders numerous protocols in quantum information theory computationally
tractable and is an important new efficient quantum circuit family which goes
significantly beyond the standard paradigm of the quantum Fourier transform.
