  A conjecture due to Bohigas, Giannoni and Schmit (BGS), stating that the
energy level correlations of quantum chaotic systems generically obey the laws
of random matrix theory, is given a precise formulation for quantized
symplectic maps. No statement is made about any individual quantum map. Rather,
a few-parameter ensemble of maps is considered, such that the deterministic map
is composed with a diffusion operator on average. The ensemble is a ``quantum''
one, which is to say that the diffusion operator contracts to the identity in
the classical limit. It is argued that the BGS conjecture is true on average
over such an ensemble, provided that the classical map is mixing. The method
used is closely related to the supersymmetric formalism of Andreev et al for
chaotic Hamiltonian systems.
