Quantum Signal Processing with the one-dimensional quantum Ising model

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.