  It is widely recognized that the main difficulty in designing devices which
could process information using quantum states is due to the decoherence of
local excitations about a ground state. A solution to this problem was
suggested in \cite{Kitaev}, relying on (non-local) topological excitations,
structurally protected against local noise. However, a practical implementation
of this proposal using special Landau levels in fractional quantum Hall effect
systems (FQHE) \cite{QHE} has proven elusive, while accessible FQHE states are
theoretically not optimal because their representations in the Hilbert space of
states are not dense. We propose using a different physical system (cold Fermi
atoms), whose semiclassical dynamics is described by a hyperelliptic function
in the Sklyanin formalism. The homological structure of the complex curve
corresponds to representations of the braid group, with the action of Hecke
operators leading to singularities detectable in the semiclassical
oscillations. We argue that, for a fixed genus of the hyperelliptic curve, the
Richardson-Gaudin pairing Hamiltonian problem is the singular limit $m \to
\infty$ of level-$k$ $\widehat{sl}(2)$, with $k+2 = \frac{4}{8m+1}\to 0$, so
that the level $k$ is admissible in the sense of Kac and Kazhdan \cite{KK}, but
the corresponding Hecke algebra is a $q-$deformation of the symmetric group
with fixed $q = e^{i\pi/4}$, as $m \to \infty$.
