QEC Code Strategies
===================

The landscape of quantum error correction code architectures.

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   * - **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 :math:`\\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 :math:`\\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
                    \(\|\alpha\rangle \pm \|-\alpha\rangle\). 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 
                    :math:`\\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.