Fast quantum algorithms for approximating the irreducible representations of groups

All representations of finite groups and compact linear groups can be expressed as unitary matrices given an appropriate choice of basis. This makes them natural candidates for implementation using quantum circuits. As shown here, the irreducible representations of the symmetric group Sn, the alternating group An, the unitary groups U(n) and SU(n), and the special orthogonal group SO(n) can each be efficiently implemented by quantum circuits. Using the Hadamard test one can thus approximate individual matrix elements of the irreducible representations of these groups to within ±ǫ in time polynomial in n and 1/ǫ on a quantum computer. I am aware of no polynomial-time classical algorithm that achieves this.