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

Abstract

Quantum signal processing (QSP) transforms a unitary parametrized by a classical variable θ into one governed by a polynomial function f ( θ ) . Though quantum mechanics is linear, such highly nonlinear transformations arise naturally from the curvature of the qubit Bloch sphere. This capability underpins many quantum algorithms and enables robust control by decreasing sensitivity to errors in a control parameter, or quantum sensing by increasing sensitivity to variations one seeks to measure. Here, we extend QSP from classical, commuting control parameters to noncommuting quantum operators. We call this new multivariate class . Rather than treating the noncommutativity of quantum control variables as a complication, non-Abelian QSP exploits their richer commutator algebra as a resource for efficient control. A natural setting is hybrid oscillator-qubit systems realized in superconducting and trapped-ion processors, where the control variables can be the noncommuting position and momentum of a quantum harmonic oscillator. We introduce the Gaussian-controlled rotation (GCR), a canonical instance of non-Abelian QSP that cancels the Gaussian quantum-fluctuation error of an oscillator-controlled qubit rotation, matching the robustness of classical composite pulses with a much shorter circuit and an order-of-magnitude lower error. GCR enables fully analytical circuits for preparing squeezed, cat, GKP, and Fock states with performance comparable to state-of-the-art numerically optimized protocols. Building on this primitive, we develop an analytical framework for universal control of GKP bosonic codes, in which logical readout doubles as piecewise gate teleportation, yielding the first high-fidelity, error-mitigated single- and two-qubit logical gates for finite-energy GKP that surpass prior schemes even in the noiseless regime. The same framework supplies midcircuit error detection and extends in closed form to arbitrary lattices, qudits, and multimode codes relevant to the superconducting-circuit experiments now demonstrating beyond-break-even GKP memory. We further apply these ideas to close a key gap in oscillator-assisted quantum phase estimation. Together, these results show that the non-Abelian structure of quantum control operators can provide a practical resource for robust and efficient quantum control, establishing non-Abelian QSP as both a new theoretical framework and an experimentally accessible control paradigm.