This thesis introduces and analyses a new model of time-periodic (Floquet) dynamics
in a quantum spin systems. This model is implemented via a time-periodic quantum
circuit with local Clifford gates. All the results of this thesis are rigorous mathematical proofs, which use tools and methods from quantum information science
to study problems in many-body quantum systems and condensed-matter physics.
This includes proofs of a form of dynamical mixing of Pauli operators in the case of
local interactions, and conditions under which the evolution operator can resemble a
random unitary. The scrambling time is of critical importance to these results, and in
the case of non-local interactions, a slightly larger than logarithmic scrambling time
is found. Also, the model analysed in this thesis has the peculiarity that it displays a
strong form of localisation in one spatial dimension and the absence of localisation in
two dimensions. There is no previously known model with these features, hence, this
research is important to characterise the landscape of many-body quantum physics
