Abstract
A system of quantum computing structures is introduced and proven capable of making emerge, on average, the orbits of classical bounded nonlinear maps on \mathbb\{C\} through the iterative action of path-dependent quantum gates. The effects of emerging nonlinear dynamics and chaos upon the quantum averages of relevant observables and quantum probabilities are exemplified for a version of Chirikov's standard map on \mathbb\{C\} . Both the individual orbits and ensemble properties are addressed so that the Poincar\'e map for Chirikov's standard map, in the current quantum setting, is reinterpreted in terms of a quantum ensemble which is then formally introduced within the formalized system of quantum computing structures, in terms of quantum register machines, revealing three phases of quantum ensemble dynamics: the regular, the chaotic and an intermediate phase called complex quantum stochastic phase which shares similarities to the edge of chaos notion from classical cellular automata