  The quantum Schur transform is a fundamental building block that maps the
computational basis to a coupled basis consisting of irreducible
representations of the unitary and symmetric groups. Equivalently, it may be
regarded as a change of basis from the computational basis to a simultaneous
spin eigenbasis of Permutational Quantum Computing (PQC) [Quantum Inf. Comput.,
10, 470-497 (2010)]. By adopting the latter perspective, we present a
transparent algorithm for implementing the qubit quantum Schur transform which
uses $O(\log(n))$ ancillas and can be decomposed into a sequence of
$O(n^3\log(n)\log(\frac{n}{\epsilon}))$ Clifford + T gates, where $\epsilon$ is
the accuracy of the algorithm in terms of the trace norm. We discuss the
necessity for some applications of implementing this operation as a unitary
rather than an isometry, as is often presented.
  By studying the associated Schur states, which consist of qubits coupled via
Clebsch-Gordan coefficients, we introduce the notion of generally coupled
quantum states. We present six conditions, which in different combinations
ensure the efficient preparation of these states on a quantum computer or their
classical simulability (in the sense of computational tractability). It is
shown that Wigner 6-j symbols and SU(N) Clebsch-Gordan coefficients naturally
fit our framework. Finally, we investigate unitary transformations which
preserve the class of computationally tractable states.
