Non-Abelian Anyons and Topological Phases: Quantum Computation and Quantum Simulation

Abstract

A central challenge at the intersection of quantum condensed-matter physics and quantum information science is to understand how exotic topological phases and quantum resources govern the power of quantum computation and simulation---and to translate that understanding into practical architectures. This thesis contributes to this program along three interconnected directions: enabling fault-tolerant universal quantum computation with non-Abelian topological order, establishing the fundamental computational reach of near-term analog quantum simulators, and delineating the boundary between classical and quantum computational power through the lenses of topology and magic. In the first direction, drawing on the structure of non-Abelian topological order, we address the dominant resource overhead in current quantum computing architectures---magic-state distillation---by moving beyond the standard stabilizer formalism entirely. We construct a comprehensive framework for fault-tolerant universal quantum computation based on the non-Abelian quantum double model $\mathcal{D}(S_3)$, with all operations realized on qubit--qutrit degrees of freedom in a planar, geometrically local architecture feasible on all current experimental platforms. This work addresses the long-standing open question of realizable, fault-tolerant universal topological quantum computation. In the second direction, we establish the computational universality of analog quantum simulators operating under global control---platforms that circumvent the demanding requirements of digitization and individual addressability. We prove a necessary and sufficient condition for universality in globally controlled qubit chains, showing that widely used platforms such as Rydberg atom arrays and trapped ions are generically universal once lattice reflection symmetry is broken. We further derive sufficient conditions for universality in fermionic and bosonic optical superlattices. As an experimental demonstration of this framework, we realize the dynamics of the cluster-state $ZXZ$ model---an exemplar of symmetry-protected topological order---in Rydberg atom arrays, validating the theoretical foundation. In the third direction, we sharpen the classical--quantum computational boundary from both sides. We show that non-Abelian Majorana braiding realized in classical mechanical metamaterials admits a faithful classical simulation, establishing that braiding-only architectures are classically simulable without supplementary non-Clifford resources. Conversely, we prove that random matrix product states carry magic growing exponentially with system size, demonstrating that generic one-dimensional many-body states harbor ample non-stabilizer resources. Together, these results delineate precisely where genuine quantum advantage emerges in topological and many-body systems. Taken as a whole, this thesis leverages the convergence of topological matter and quantum information to remove key bottlenecks toward fault-tolerant quantum computation, to place near-term quantum simulation on rigorous footing, and to clarify the fundamental resources that separate classical from quantum computational power.