  Quantum integrability has proven to be a useful tool to study quantum
many-body systems out of equilibrium. In this paper we construct a generic
framework for integrable quantum circuits through the procedure of Floquet
Baxterisation. The integrability is guaranteed by establishing a connection
between Floquet evolution operators and inhomogeneous transfer matrices
obtained from the Yang-Baxter relations. This allows us to construct integrable
Floquet evolution operators with arbitrary depths and various boundary
conditions. Furthermore, we focus on the example related to the staggered
6-vertex model. In the scaling limit we establish a connection of this Floquet
protocol with a non-rational conformal field theory. Employing the properties
of the underlying affine Temperley--Lieb algebraic structure, we demonstrate
the dynamical anti-unitary symmetry breaking in the easy-plane regime. We also
give an overview of integrability-related quantum circuits, highlighting future
research directions.
