QEC Code Strategies

The landscape of quantum error correction code architectures.

Shor Code	First quantum error correction code. Protects against both bit-flip and phase-flip errors. Proved that quantum errors can be corrected. (1) logical qubit (= 9) physical qubits.
Steane Code	More efficient than the Shor Code. Uses fewer qubits, making some logical operations easier. Inspired many modern quantum codes. (1) logical qubit (= 7) physical qubits.
Toric Code	First topological quantum error correction code. Stores quantum information across a lattice structure using topological properties. (1) logical qubit is encoded across a 2D lattice.
Surface Code	Practical implementation of topological QEC. Most widely used QEC code today, compatible with current 2D grids, and a strong foundation for scalable fault-tolerant roadmaps. (1) logical qubit \(\\sim 10^2\\text{--}10^3+\) physical qubits.
Color Code	Topological QEC code using a colored lattice. Supports more transversal logical gates than Surface Codes, simplifying certain fault-tolerant operations. (1) logical qubit \(\\sim 10^2\\text{--}10^3+\) physical qubits.
Bacon-Shor Code	Subsystem QEC code. Combines features of Shor Code and surface-like structures. Requires fewer stabilizer measurements, simplifying error detection. (1) logical qubit is encoded using a grid of physical qubits.
GKP Code	Continuous-variable QEC code. Encodes a logical qubit in the states of a quantum oscillator. Corrects small displacement errors. Highly efficient for bosonic hardware: (1) logical qubit (= 1) oscillator mode.
Cat Code	Bosonic QEC code. Encodes information using superpositions of coherent states (|alpharangle pm |-alpharangle). Protects against photon loss.
XZZX Surface Code	Advanced version of the Surface Code using a different arrangement of (X) and (Z) stabilizers. Performs better under biased noise. (1) logical qubit \(\\sim 10^2\\text{--}10^3+\) physical qubits.
Hypergraph Product (LDPC) Code	Quantum Low-Density Parity-Check (LDPC) code. Reduces the number of physical qubits required for error correction, enabling more efficient encoding. (1) logical qubit has lower qubit overhead than Surface Codes.