  Quantum Signal Processing (QSP) has emerged as a promising framework to
manipulate and determine properties of quantum systems. QSP not only unifies
most existing quantum algorithms but also provides tools to discover new ones.
Quantum signal processing is applicable to single- or multi-qubit systems that
can be qubitized so one can exploit the SU$(2)$ structure of system evolution
within special invariant two-dimensional subspaces. In the context of quantum
algorithms, this SU$(2)$ structure is artificially imposed on the system
through highly nonlocal evolution operators that are difficult to implement on
near-term quantum devices. In this work, we propose QSP protocols for the
infinite-dimensional Onsager Lie Algebra, which is relevant to the physical
dynamics of quantum devices that can simulate the transverse field Ising model.
To this end, we consider QSP sequences in the Heisenberg picture, allowing us
to exploit the emergent SU$(2)$ structure in momentum space and synthesize QSP
sequences for the Onsager algebra. Our results demonstrate a concrete
connection between QSP techniques and Noisy Intermediate Scale quantum
protocols. We provide examples and applications of our approach in diverse
fields ranging from space-time dual quantum circuits and quantum simulation, to
quantum control.
