Towards Non-Abelian Quantum Signal Processing: Efficient Control of Hybrid Continuous- and Discrete-Variable Architectures

Robust quantum control is crucial for achieving operations below the quantum error correction threshold. Quantum Signal Processing (QSP) transforms a unitary parameterized by $\theta$ into one governed by a polynomial function $f(\theta)$, a feature that underpins key quantum algorithms. Originating from composite pulse techniques in NMR, QSP enhances robustness against systematic control errors. We extend QSP to a new class, non-abelian QSP, which utilizes non-commuting control parameters, $\hat\theta_1, \hat\theta_2, \dots$, representing quantum harmonic oscillator positions and momenta. We introduce a fundamental non-abelian composite pulse sequence, the Gaussian-Controlled-Rotation (GCR), for entangling and disentangling a qubit from an oscillator. This sequence achieves at least a $4.5\times$ speedup compared to the state-of-the-art abelian QSP pulse BB1, while maintaining performance. Though quantum fluctuations in the control parameters are unavoidable, the richer commutator algebra of non-abelian QSP enhances its power and efficiency. Non-abelian QSP represents the highest tier of QSP variants tailored for hybrid oscillator-qubit architectures, unlocking new possibilities for such systems. We demonstrate the utility of GCR in high-fidelity preparation of continuous-variable oscillator states, including squeezed, Fock, cat, and GKP states, using fully analytical schemes that match numerically optimized methods in fidelity and depth while enabling mid-circuit error detection. Furthermore, we propose a high-fidelity QSP-based end-of-the-line GKP readout and a measurement-free, error-corrected gate teleportation protocol for logical operations on GKP bosonic qudits, bridging the gap between idealized theoretical and experimentally realistic versions of the GKP code. Finally, we showcase a GCR-based phase estimation algorithm for oscillator-based quantum computing.