Topological quantum codes and memories：from error chains and thresholds to self-correction and quantum low-density parity-check codes

Quantum information cannot be copied arbitrarily and is vulnerable to bitflip, phase-flip, and measurement errors, making the preservation of an unknown quantum state fundamentally more demanding than the storage of classical information. Topological quantum error correction protects logical information by distributing it nonlocally across many physical qubits, so that local faults produce measurable error syndromes. A decoder uses these syndromes to infer an appropriate recovery operation while avoiding system-spanning error chains that implement nontrivial logical operators. The error-correction threshold identifies the regime in which increasing the code distance leads to a sustained reduction in the logical error rate and therefore provides a basic criterion for scalable quantum memories. Starting from an intuitive description of stabilizers, error chains, and homology, we present a review of active error correction with surface codes and recent experimental progress. We then discuss how low-dimensional locality places fundamental constraints on coding overhead, motivating two complementary directions: passive self-correcting quantum memories and good quantum low-density parity-check codes.