Quantum Error Correction

Quantum Error Correction (QEC)

QEC protects fragile quantum information from decoherence and environmental noise using logical qubits encoded across multiple physical qubits.

Surface Codes and Topological Qubits

Surface codes are a class of stabilizer codes mapped onto a 2D lattice.

• Qubits are placed on the edges of the lattice.
• Star (vertex) operators $A_v = \prod_{i \in \text{star}} X_i$ and plaquette operators $B_p = \prod_{i \in \text{boundary}} Z_i$ form the stabilizer generators.

Syndrome Measurement

Error detection occurs without measuring the logical state via syndrome extraction circuits using ancillary measure qubits.

flowchart TD
%%{init: {"theme": "default", "flowchart": {"useMaxWidth": true}}}%%
    subgraph SyndromeExtractionSyndromeExtraction ["Syndrome Extraction<br><br><br>"]
        Data[Data Qubits] -->|"CNOT(Data, Ancilla)"| Ancilla[Measure Qubit]
        Ancilla -->|"Measure(Z)"| Syndrome[Syndrome Graph]
    end
    subgraph DecoderMWPMDecoder ["MWPM Decoder<br><br><br>"]
        Syndrome -->|"MapToGraph()"| Match[Minimum Weight Perfect Matching]
        Match -->|"ApplyCorrection()"| Recovery[Recovery Operations]
    end

Decoder Topology

The Minimum Weight Perfect Matching (MWPM) algorithm pairs any detected error defects (anyons) in the space-time syndrome graph.