  We consider a quaternately generalized Pfaffian
QGPf$(\frac{1}{J(z_i,z_j,z_k,z_l)})[J(z_1,...,z_N)]^2$ in which the square of
Vandermonde determinant, $[J(z_1,...,z_N)]^2$, implies the upmost Landau level
is half filled. This wave function is the unique highest density zero energy
state of a special short range interacting Hamiltonian. One can think this
quaternate composite fermion liquid as a competing ground state of Moore-Read
(MR) Pfaffian state at $\nu=5/2$. The degeneracy of the quasihole excitations
above the QGPf is higher than that of Moore-Read even Read-Rezayi quasiholes.
The QGPf is related to a unitary conformal field theory with $Z_2\times Z_2$
parafermions in coset space $SU(3)_2/U(1)^2$ . Because of the level-rank
duality between $SU(3)_2$ and $SU(2)_3$ in conformal field theory, these
quasiholes above this QGPf state obeying non-abelian anyonic statistics are
expected to support the universal quantum computation at $\nu=5/2$ as
Read-Rezayi quasiholes at $\nu=13/5$. The edge states of QGPf are very
different from those of the Pfaffian's.
