What are the methods used for quantum error correction, and how do they work?

Quantum error correction (QEC) uses specialized encoding and measurement techniques to detect and correct errors in qubits caused by noise. Three widely studied methods are the Shor code, stabilizer codes (like the Steane code), and topological codes (like the surface code). Each approach encodes logical qubits into multiple physical qubits and uses redundancy to identify errors without directly measuring the quantum state, which would collapse its superposition.

The Shor code, one of the first QEC methods, encodes a single logical qubit into nine physical qubits. It combines protection against bit-flip (X) and phase-flip (Z) errors. First, the logical qubit is split into three qubits using a repetition code for bit-flip detection. Each of those three is then encoded into another trio using a phase-flip code. To detect errors, ancilla qubits are entangled with the encoded block to measure "syndromes"—patterns that reveal which qubit flipped. For example, measuring the parity (difference) between pairs of qubits highlights a bit-flip, while similar phase parity checks identify phase-flips. Once the syndrome is known, a corrective operation is applied. While effective, the Shor code’s high qubit overhead makes it impractical for large-scale systems.

Stabilizer codes, such as the Steane code, use mathematical symmetries (stabilizers) to detect errors. The Steane code encodes one logical qubit into seven physical qubits. Stabilizers are operators (e.g., combinations of Pauli X and Z gates) that leave valid code states unchanged. By measuring these operators, you get syndromes without disturbing the encoded data. For instance, measuring X-type stabilizers detects Z errors (phase-flips), while Z-type stabilizers detect X errors (bit-flips). The Steane code’s stabilizers are designed similarly to classical Hamming codes, enabling correction of single-qubit errors. This method is more efficient than the Shor code but still requires precise measurement and low physical error rates to function reliably.

The surface code is a topological QEC scheme that arranges qubits in a 2D lattice, where each qubit interacts with its nearest neighbors. Logical qubits are encoded in the collective state of many physical qubits, and errors are detected by measuring stabilizers defined on plaquettes (squares) of the lattice. For example, alternating plaquettes measure X-type (checking for Z errors) and Z-type (checking for X errors) stabilizers. Errors create detectable “chains” of syndrome changes, and the shortest path between these chains determines the correction. The surface code’s key advantage is its high fault tolerance and scalability—it can tolerate higher physical error rates and requires only local interactions, making it a leading candidate for practical quantum computers. However, it still demands thousands of physical qubits per logical qubit, highlighting the challenge of achieving useful quantum advantage.