  In recent years there has been an increasing interest on the theoretical and
experimental investigation of space-time dual quantum circuits. They exhibit
unique properties and have applications to diverse fields. Periodic space-time
dual quantum circuits are of special interest, due to their iterative structure
defined by the Floquet operator. A very similar iterative structure naturally
appears in Quantum Signal processing (QSP), which has emerged as a framework
that embodies all the known quantum algorithms. However, it is yet unclear
whether there is deeper relation between these two apparently different
concepts. In this work, we establish a relation between a circuit defining a
Floquet operator in a single period and its space-time dual defining QSP
sequences for the Lie algebra sl$(2,\mathbb{C})$, which is the complexification
of su$(2)$. First, we show that our complexified QSP sequences can be
interpreted in terms of action of the Lorentz group on density matrices and
that they can be interpreted as hybrid circuits involving unitaries and
measurements. We also show that unitary representations of our QSP sequences
exist, although they are infinite-dimensional and are defined for bosonic
operators in the Heisenberg picture. Finally, we also show the relation between
our complexified QSP and the nonlinear Fourier transform for
sl$(2,\mathbb{C})$, which is a generalization of the previous results on
su$(2)$ QSP.
