Surface codes are quantum error-correcting codes that utilize the topology of a two-dimensional lattice to protect quantum information against noise. They are a vital component in the development of fault-tolerant quantum computing due to their robustness and ability to efficiently detect and correct errors in quantum bits (qubits).
Error correction in quantum computing is essential due to the fragile nature of quantum states, which are prone to errors from decoherence and other quantum noise. Quantum error correction schemes, like the surface code, uniquely exploit entanglement and superposition to detect and correct errors without directly measuring the quantum states involved.
Non-Abelian anyons are exotic quasiparticles that arise in certain two-dimensional systems and have the unique property that interchanging them results in a change to the system's quantum state that depends on the order of the exchanges. This property makes non-Abelian anyons promising candidates for fault-tolerant quantum computation, as they can potentially perform operations that are inherently robust against local errors.