Non-Abelian Quantum Signal Processing: A Composite Pulse for Fast Analytic Control of Hybrid Oscillator-Qubit Processors

Quantum Signal Processing (QSP) transforms a unitary parameterized by a classical variable $\theta$ into one governed by a polynomial function $f(\theta)$. Though quantum mechanics is linear, such highly nonlinear transformations arise naturally from the curvature of the qubit Bloch sphere. The QSP primitive underpins most quantum algorithms and finds broad utility in robust control by decreasing sensitivity to parameter errors, and in quantum sensing by increasing sensitivity to target parameters. In this work, we extend QSP to a new multivariate class, non-Abelian QSP, that utilizes a set of non-commuting (operator-valued) control parameters $\{\hat{\theta}_1, \hat{\theta}_2, \dots\}$. Experimental instantiations of this richer algebraic structure are currently being explored in hybrid oscillator-qubit systems realized in superconducting and trapped-ion processors, where the non-commuting variables are oscillator positions and momenta. We demonstrate the utility of our construction, the Gaussian-controlled-rotation (GCR) which is a canonical instance of this class, across three domains: fully analytical state preparation circuits whose performance matches state-of-the-art machine-learning protocols for preparing squeezed, cat, GKP, and Fock states; a complete analytical framework for universal control of GKP bosonic error-corrected qubits, including logical readout and error-corrected gate teleportation --with mid-circuit error detection and generalization to arbitrary lattices, qudits, and multi-mode codes uniquely enabled by the analytical structure; and a construction closing a key gap in oscillator-aided quantum phase estimation algorithms. These results establish non-Abelian QSP as a powerful new frontier, one that is not merely of theoretical interest but ready to be put to work in the laboratory today.