  Floquet quantum circuits are able to realise a wide range of non-equilibrium
quantum states, exhibiting quantum chaos, topological order and localisation.
In this work, we investigate the stability of operator localisation and
emergence of chaos in random Floquet-Clifford circuits subjected to unitary
perturbations which drive them away from the Clifford limit. We construct a
nearest-neighbour Clifford circuit with a brickwork pattern and study the
effect of including disordered non-Clifford gates. The perturbations are
uniformly sampled from single-qubit unitaries with probability $p$ on each
qubit. We show that the interacting model exhibits strong localisation of
operators for $0 \le p < 1$ that is characterised by the fragmentation of
operator space into disjoint sectors due to the appearance of wall
configurations. Such walls give rise to emergent local integrals of motion for
the circuit that we construct exactly. We analytically establish the stability
of localisation against generic perturbations and calculate the average length
of operator spreading tunable by $p$. Although our circuit is not separable
across any bi-partition, we further show that the operator localisation leads
to an entanglement bottleneck, where initially unentangled states remain weakly
entangled across typical fragment boundaries. Finally, we study the spectral
form factor (SFF) to characterise the chaotic properties of the operator
fragments and spectral fluctuations as a probe of non-ergodicity. In the $p =
1$ model, the emergence of a fragmentation time scale is found before random
matrix theory sets in after which the SFF can be approximated by that of the
circular unitary ensemble. Our work provides an explicit description of quantum
phases in operator dynamics and circuit ergodicity which can be realised on
current NISQ devices.
