Towards Non-Abelian Quantum Signal Processing: Error-Corrected Universal Gate Teleportation for GKP Codes

Quantum signal processing (QSP) is a technique for preparing arbitrary unitary operations using multiple single-qubit rotations where the control paramters are classical variables. We extend the concept of quantum signal processing to the case of multiple control parameters which are themselves non-commuting quantum operators--namely the positions and momenta of quantum harmonic oscillators. The non-commutativity of the control parameters implies that they unavoidably suffer intrinsic quantum fluctuations, but the richer commutator algebra also significantly enhances the power of quantum signal processing and reduces circuit depths. Non-abelian QSP provides a strong framework with which to analyze quantum control of hybrid continuous-variable (CV) -- discrete-variable (DV) systems comprising oscillators and qubits such as those found in trapped-ion and superconducting circuit QED systems. We illustrate the power of this framework with applications to the Gottesman-Kitaev-Preskill codes. The framework's versatility bridges the gap between theoretically ideal infinitely-squeezed GKP states and experimentally realistic finitely-squeezed GKP states, significantly enhancing the fidelity of practical GKP gate operations. In particular, for this talk, we introduce an error-corrected universal single and multi-qubit logical gate teleportation scheme which corrects errors on the oscillator (GKP qubit) while teleporting the gates using a single ancilla qubit. We develop a novel piece-wise gate teleportation approach that mitigates the effects of ancilla dephasing, making ancilla bit flips the leading source of noise. Our scheme can be realized in a fault-tolerant manner using a biased-noise ancilla at no additional cost in comparison to the stabilization circuit of the GKP codes. Such construction was only possible due to the understanding of hybrid oscillator-qubit systems that we achieved from the non-abelian QSP framework.