Topological quantum computing: comprehensive overview and theoretical foundations

Abstract

The apparent computational power of Quantum computers suggests a significant speed- up in calculations by leveraging the principles of superposition and entanglement, proper of Quantum Mechanics. However, the quantum information stored is very fragile to thermal excitations (also referred to as noise) and accuracy errors when information is processed via quantum logical gates. Many research areas develop techniques to correct and mitigate such errors; however, progress on the scalability and efficiency of quantum chips is slow. On the other hand, a relatively new technology has been proposed, directly targeting the prob- lem of decoherence by an inherently fault-tolerant quantum processor, Topological quantum Computing (TQC). In this overview, we trace State of the Art, the fundamental concepts, and the theoretical framework to understand systematically the physical consequence of its constituent qubits, non-Abelian anyons, and how to perform computation. The notion of topologically ordered states is built from a toy model, the Toric code, to a physical system exhibiting topological states, the Quantum Hall Effect (QHE). Once topological order is de- rived, the connection of non-Abelian anyons as qubits is bridged by a qualitative explanation, and the comprehension of the fundamental pillars of TQC flows more naturally. Concepts are reviewed under a comprehensive perspective, using clear language, without repetitive expla- nations found in other sources, and at a technisism level readable for a profile with a strong basic Quantum Mechanics background. Finally, a temporal sequence of the most cited and important achievements on topologically ordered systems and states with the perspective on building the first topological quantum computer closes this document, offering a wide picture of today’s state of the art on Topological Quantum Computing.