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

Quantum Signal Processing with the one-dimensional quantum Ising model