Lecture 32: Classical and Quantum Error Correction: Simple Example

Classical bit-flip error correction code is described, based on redundancy and majority rule. Error suppression is evaluated, and threshold for error tolerance is determined using recursive embedding of the code. Quantum extension replaces redundancy with entanglement, extracts syndromes using ancillas, corrects both bit-flip and phase-flip errors. Fault-tolerant construction can protect the computation indefinitely, while making error rates go below the threshold is a challenge for hardware technology. Important aspects of computational complexity analysis are reemphasised: Scaling with respect to input and output sizes, uniform circuits, bounded errors, and choice between worst case and average case performance.