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.
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