Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantum Fourier Transforms and the Complexity of Link Invariants for
  Quantum Doubles of Finite Groups   Knot and link invariants naturally arise from any braided Hopf algebra. We
consider the computational complexity of the invariants arising from an
elementary family of finite-dimensional Hopf algebras: quantum doubles of
finite groups (denoted D(G), for a group G). Regarding algorithms for these
invariants, we develop quantum circuits for the quantum Fourier transform over
D(G); in general, we show that when one can uniformly and efficiently carry out
the 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 apply
these results to the symmetric groups to yield efficient circuits for the
quantum Fourier transform over D(S_n). With such a Fourier transform, it is
straightforward to obtain additive approximation algorithms for the related
link invariant. Additionally, we show that certain D(G) invariants (such as
D(A_n) invariants) are BPP-hard to additively approximate, SBP-hard to
multiplicatively approximate, and #P-hard to exactly evaluate. Finally, we make
partial progress on the question of simulating anyonic computation in groups
uniformly as a function of the group size. In this direction, we provide
efficient 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 of
G and an irrep of a centralizer, we present an efficient implementation under
certain conditions such as when there is an efficient Clebsch-Gordan transform
over the centralizers. We remark that this also provides a simulation of
certain anyonic models of quantum computation, even in circumstances where the
group may have size exponential in the size of the circuit.

14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas