  Operators in ergodic spin-chains are found to grow according to
hydrodynamical equations of motion. The study of such operator spreading has
aided our understanding of many-body quantum chaos in spin-chains. Here we
initiate the study of "operator spreading" in quantum maps on a torus, systems
which do not have a tensor-product Hilbert space or a notion of spatial
locality. Using the perturbed Arnold cat map as an example, we analytically
compare and contrast the evolutions of functions on classical phase space and
quantum operator evolutions, and identify distinct timescales that characterize
the dynamics of operators in quantum chaotic maps. Until an Ehrenfest time, the
quantum system exhibits classical chaos, i.e. it mimics the behavior of the
corresponding classical system. After an operator scrambling time, the operator
looks "random" in the initial basis, a characteristic feature of quantum chaos.
These timescales can be related to the quasi-energy spectrum of the unitary via
the spectral form factor. Furthermore, we show examples of "emergent
classicality" in quantum problems far away from the classical limit. Finally,
we study operator evolution in non-chaotic and mixed quantum maps using the
Chirikov standard map as an example.
