The paper by Dennis, Kitaev, Landahl, and Preskill analyzes surface codes, a family of topological quantum error-correcting codes introduced by Kitaev. These codes arrange qubits in a two-dimensional array on a nontrivial topological surface, with encoded quantum operations associated with nontrivial homology cycles. The authors formulate protocols for error recovery and study their efficacy, identifying an order-disorder phase transition at a nonzero critical error rate, known as the accuracy threshold. This transition can be modeled by a three-dimensional $Z_2$ lattice gauge theory with quenched disorder. They estimate the accuracy threshold under assumptions of local quantum gates, rapid qubit measurements, and polynomial-size classical computations. A robust recovery procedure is devised that does not require measurement or fast classical processing, but it relies on local quantum gates in four or more spatial dimensions. The paper discusses encoding, measurement, and fault-tolerant universal quantum computation using surface codes, arguing that these codes offer a promising framework for quantum computing architectures. The authors also explore the physical realization of surface codes as intrinsically stable quantum memories, where decoherence is resisted without active information processing.The paper by Dennis, Kitaev, Landahl, and Preskill analyzes surface codes, a family of topological quantum error-correcting codes introduced by Kitaev. These codes arrange qubits in a two-dimensional array on a nontrivial topological surface, with encoded quantum operations associated with nontrivial homology cycles. The authors formulate protocols for error recovery and study their efficacy, identifying an order-disorder phase transition at a nonzero critical error rate, known as the accuracy threshold. This transition can be modeled by a three-dimensional $Z_2$ lattice gauge theory with quenched disorder. They estimate the accuracy threshold under assumptions of local quantum gates, rapid qubit measurements, and polynomial-size classical computations. A robust recovery procedure is devised that does not require measurement or fast classical processing, but it relies on local quantum gates in four or more spatial dimensions. The paper discusses encoding, measurement, and fault-tolerant universal quantum computation using surface codes, arguing that these codes offer a promising framework for quantum computing architectures. The authors also explore the physical realization of surface codes as intrinsically stable quantum memories, where decoherence is resisted without active information processing.