QEC

Quantum error correction (QEC) is the set of methods for protecting quantum information from errors due to decoherence, noise, and imperfect operations. Because quantum information cannot be copied, QEC uses entanglement and redundancy across multiple physical qubits to encode a logical qubit.

QEC encodes quantum information into multiple physical qubits, performs syndrome measurements, and applies corrections. Quantum errors are typically modeled as X (bit-flip), Z (phase-flip), or Y errors. Stabilizer codes use commuting observables to detect error syndromes without measuring the logical state, enabling recovery. Fault-tolerant designs limit error propagation during syndrome extraction.

Prominent families include Shor's 9-qubit code and Steane's 7-qubit code (CSS codes), concatenated codes, and topological codes such as the surface code and color codes. Surface codes, implemented on a two-dimensional lattice with nearest-neighbor interactions, are among the most studied due to high error thresholds and practical locality.

Theoretical foundations were developed in the 1990s, culminating in the threshold theorem: if physical error rates are below a certain value, scalable fault-tolerant quantum computation is possible with overhead that grows polynomially with computation size. Key concepts include stabilizer formalism and logical qubits.

Experimentally, small-scale demonstrations have shown error detection and correction on a few qubits, and ongoing work aims to realize larger surface codes with acceptable qubit overhead. Challenges include achieving low physical error rates, reliable syndrome measurements, and reducing resource overhead.

QEC is essential for reliable quantum memories and fault-tolerant quantum computing, enabling long computations and robust quantum communication in the presence of noise.