Robust non-Abelian geometric phases on three-qubit spin codes

Abstract: Quantum holonomies are non-Abelian Geometric Phases predominantly observed in adiabatic, non-adiabatic, or measurement-based quantum evolutions. Their significance lies in their potential utility within quantum computing due to their robustness against noise throughout the parameter path. In this report, we detail the foundational methods necessary for constructing holonomic non-Abelian gates specifically designed for tripartite states and , which serve as the logical qubits in our project. Given that the existence of a universal set of gates has already been demonstrated for each of these evolution types, our project delves into the advantages of applying these basis states across the three evolution categories. We have reformulated the Nuclear Quadrupole Resonance (NQR) Hamiltonian to be exclusively composed of two-body terms, thus rendering it more experimentally feasible. Furthermore, we have connected the W states with the remaining tripartite states to construct a four-level model system and generalized gates within this framework. Lastly, we introduce a measurement-based method that maintains its non-Abelian attributes even in the Zeno limit, where the process of projective measurement gradually approaches the adiabatic model.