This paper presents an analysis of surface codes, a type of topological quantum error-correcting code introduced by Kitaev. These codes use qubits arranged in a two-dimensional lattice on a surface with nontrivial topology, and quantum operations are associated with homology cycles of the surface. The paper discusses protocols for error recovery and the efficacy of these protocols, showing that an order-disorder phase transition occurs at a nonzero critical error rate. If the error rate is below this threshold, encoded information can be protected arbitrarily well. The phase transition is modeled by a three-dimensional Z₂ lattice gauge theory with quenched disorder. The paper estimates the accuracy threshold, assuming local quantum gates, rapid measurements, and instantaneous classical computations. It also proposes a robust recovery procedure that does not require measurement or fast classical processing, though it requires local gates only if qubits are arranged in four or more spatial dimensions. The paper discusses procedures for encoding, measuring, and performing fault-tolerant universal quantum computation with surface codes, arguing that these codes provide a promising framework for quantum computing architectures. It emphasizes the importance of fault tolerance in quantum computing, noting that quantum states can be encoded to resist decoherence and that fault-tolerant protocols allow reliable processing of encoded quantum states with imperfect components. The paper also discusses the threshold theorem, which states that arbitrarily long quantum computations can be executed with high reliability if the error rate of quantum gates is below a certain critical value. The paper estimates this critical error rate as p_c ≳ 10⁻⁴, indicating that robust quantum computation is possible if the decoherence time of stored qubits is at least 10⁴ times longer than the time needed to execute a fundamental quantum gate. The paper analyzes the accuracy threshold for quantum storage and computation, showing that the threshold for quantum computation is not as easy to analyze definitively, but its numerical value is unlikely to be substantially different. The paper also discusses the design of a fault-tolerant quantum computer incorporating surface codes, emphasizing the importance of fault tolerance in quantum computing architectures. It concludes that surface codes provide a promising framework for quantum computing, with the potential to realize intrinsically stable quantum memories. The paper also discusses the properties of surface codes, including their ability to protect quantum information from errors and their potential for fault-tolerant quantum computation. The paper concludes that surface codes are a promising approach to quantum computing, with the potential to realize robust quantum memories and perform fault-tolerant quantum computation.This paper presents an analysis of surface codes, a type of topological quantum error-correcting code introduced by Kitaev. These codes use qubits arranged in a two-dimensional lattice on a surface with nontrivial topology, and quantum operations are associated with homology cycles of the surface. The paper discusses protocols for error recovery and the efficacy of these protocols, showing that an order-disorder phase transition occurs at a nonzero critical error rate. If the error rate is below this threshold, encoded information can be protected arbitrarily well. The phase transition is modeled by a three-dimensional Z₂ lattice gauge theory with quenched disorder. The paper estimates the accuracy threshold, assuming local quantum gates, rapid measurements, and instantaneous classical computations. It also proposes a robust recovery procedure that does not require measurement or fast classical processing, though it requires local gates only if qubits are arranged in four or more spatial dimensions. The paper discusses procedures for encoding, measuring, and performing fault-tolerant universal quantum computation with surface codes, arguing that these codes provide a promising framework for quantum computing architectures. It emphasizes the importance of fault tolerance in quantum computing, noting that quantum states can be encoded to resist decoherence and that fault-tolerant protocols allow reliable processing of encoded quantum states with imperfect components. The paper also discusses the threshold theorem, which states that arbitrarily long quantum computations can be executed with high reliability if the error rate of quantum gates is below a certain critical value. The paper estimates this critical error rate as p_c ≳ 10⁻⁴, indicating that robust quantum computation is possible if the decoherence time of stored qubits is at least 10⁴ times longer than the time needed to execute a fundamental quantum gate. The paper analyzes the accuracy threshold for quantum storage and computation, showing that the threshold for quantum computation is not as easy to analyze definitively, but its numerical value is unlikely to be substantially different. The paper also discusses the design of a fault-tolerant quantum computer incorporating surface codes, emphasizing the importance of fault tolerance in quantum computing architectures. It concludes that surface codes provide a promising framework for quantum computing, with the potential to realize intrinsically stable quantum memories. The paper also discusses the properties of surface codes, including their ability to protect quantum information from errors and their potential for fault-tolerant quantum computation. The paper concludes that surface codes are a promising approach to quantum computing, with the potential to realize robust quantum memories and perform fault-tolerant quantum computation.