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Knot and link invariants naturally arise from any braided Hopf algebra. Weconsider the computational complexity of the invariants arising from anelementary family of finite-dimensional Hopf algebras: quantum doubles offinite groups (denoted D(G), for a group G). Regarding algorithms for theseinvariants, we develop quantum circuits for the quantum Fourier transform overD(G); in general, we show that when one can uniformly and efficiently carry outthe quantum Fourier transform over the centralizers Z(g) of the elements of G,one can efficiently carry out the quantum Fourier transform over D(G). We applythese results to the symmetric groups to yield efficient circuits for thequantum Fourier transform over D(S_n). With such a Fourier transform, it isstraightforward to obtain additive approximation algorithms for the relatedlink invariant. Additionally, we show that certain D(G) invariants (such asD(A_n) invariants) are BPP-hard to additively approximate, SBP-hard tomultiplicatively approximate, and #P-hard to exactly evaluate. Finally, we makepartial progress on the question of simulating anyonic computation in groupsuniformly as a function of the group size. In this direction, we provideefficient quantum circuits for the Clebsch-Gordan transform over D(G) for"fluxon" irreps, i.e., irreps of D(G) characterized by a conjugacy class of G.For general irreps, i.e., those which are associated with a conjugacy class ofG and an irrep of a centralizer, we present an efficient implementation undercertain conditions such as when there is an efficient Clebsch-Gordan transformover the centralizers. We remark that this also provides a simulation ofcertain anyonic models of quantum computation, even in circumstances where thegroup may have size exponential in the size of the circuit.

Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups