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A conjecture due to Bohigas, Giannoni and Schmit (BGS), stating that theenergy level correlations of quantum chaotic systems generically obey the lawsof random matrix theory, is given a precise formulation for quantizedsymplectic maps. No statement is made about any individual quantum map. Rather,a few-parameter ensemble of maps is considered, such that the deterministic mapis 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 inthe classical limit. It is argued that the BGS conjecture is true on averageover such an ensemble, provided that the classical map is mixing. The methodused is closely related to the supersymmetric formalism of Andreev et al forchaotic Hamiltonian systems.

Pair Correlations of Quantum Chaotic Maps from Supersymmetry