Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform

Understand

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.