  In this paper, we study dynamical quantum networks which evolve according to
Schr\"odinger equations but subject to sequential local or global quantum
measurements. A network of qubits forms a composite quantum system whose state
undergoes unitary evolution in between periodic measurements, leading to hybrid
quantum dynamics with random jumps at discrete time instances along a
continuous orbit. The measurements either act on the entire network of qubits,
or only a subset of qubits. First of all, we reveal that this type of hybrid
quantum dynamics induces probabilistic Boolean recursions representing the
measurement outcomes. With global measurements, it is shown that such resulting
Boolean recursions define Markov chains whose state-transitions are fully
determined by the network Hamiltonian and the measurement observables.
Particularly, we establish an explicit and algebraic representation of the
underlying recursive random mapping driving such induced Markov chains. Next,
with local measurements, the resulting probabilistic Boolean dynamics is shown
to be no longer Markovian. The state transition probability at any given time
becomes dependent on the entire history of the sample path, for which we
establish a recursive way of computing such non-Markovian probability
transitions. Finally, we adopt the classical bilinear control model for the
continuous Schr\"odinger evolution, and show how the measurements affect the
controllability of the quantum networks.
