•An operatorial approach to the cellular automaton game of life is applied.•A rigorous theoretical framework is defined in order to apply the (H, ρ)-induced dynamics.•Heisenberg picture is used to evolve the system before a set of rules is applied.•Spectral analysis is applied to our quantum variation of the game of life.•The differences between the game of life and our quantum variation are highlighted. We propose an extended version of quantum dynamics for a certain system S, whose evolution is ruled by a Hamiltonian H, its initial conditions, and a suitable set ρ of rules, acting repeatedly on S. The resulting dynamics is not necessarily periodic or quasi-periodic, as one could imagine for conservative systems with a finite number of degrees of freedom. In fact, it may have quite different behaviors depending on the explicit forms of H, ρ as well as on the initial conditions. After a general discussion on this (H, ρ)-induced dynamics, we apply our general ideas to extend the classical game of life, and we analyze several aspects of this extension.