Transversal non-Clifford logical gates for quantum error-correcting codes

Abstract

Fault-tolerant quantum computation requires quantum error-correcting codes that both protect logical information and support a useful logical gate set. A central obstacle is the implementation of logical non-Clifford gates, which are necessary for universality but are not generally available through transversal operations alone. This dissertation studies an alternative approach in which coherent physical rotations, syndrome measurements, and decoding combine to generate useful logical operations on encoded qubits. It first shows that three-dimensional topological stabilizer codes can be tailored to biased Pauli noise through suitable Clifford deformations, leading to improved thresholds and subthreshold performance. It next develops a framework in which coherent physical errors followed by syndrome measurement and correction induce ensembles of logical unitaries, and shows that for surface codes these ensembles approach unitary designs above a finite threshold. Building on this perspective, the dissertation identifies a robust phase of continuous logical rotations in the surface code, in which the induced logical unitary is continuously tunable and its logical dephasing is exponentially suppressed with code distance relative to the logical rotation angle. This result motivates an adaptive protocol for implementing target logical rotations using repeated rounds of transversal rotations, syndrome extraction, and feedforward. Finally, the dissertation reports an experimental demonstration of continuous-angle logical Z rotations in the [[7,1,3]] Steane code on the IonQ Forte trapped-ion processor. Using logical Ramsey interferometry and process tomography, the resulting syndrome-resolved logical channels are characterized and shown to be well described by a dephasing model. Taken together, these results establish coherent transversal rotations and standard quantum error-correction primitives as a promising route toward continuously tunable logical non-Clifford control in quantum error-correcting codes.