Decoders!

Quantum Computers, even more so than classical ones, are sensitive to the environment around them and can experience errors. To ensure information is stored correctly and computations are carried out reliability, we must employ Error Correction.

Review: Stabilizer Formalism

Minimum Weight Perfect Matching (MWPM)

For the Toric Code, errors occur as pairs of excitations created by X or Z gates. Further, since the pauli matrices square to the identity, we can get rid of an X or Z error by simply applying the same gate. So, the problem of decoding and finding corrections boils down to pairing up errors in such a way that we can apply the minimum number of X and Z gates to get rid of them.

Union-Find

Some interesting links

• QUEKUF: An FPGA Union Find Decoder for Quantum Error Correction on the Toric Code
• Union-Find Decoder
• Recent (2025) improvements to MPWM
• Stackoverflow: Are there open source implementations of quantum error correction decoders?
• Higgott et. al. Belief-matching and Belief-find decoders
• PyMatching2: Fast Python/C++ library for decoding using MPWM