Non-binary codes
Finite-field constructions and joint decoding for qubit quantum LDPC codes.
From finite fields to qubit codes
We assign nonzero finite-field coefficients to sparse check matrices while preserving CSS commutation, then replace the coefficients by binary multiplication matrices. The resulting checks define a code on physical qubits. The finite-field representation also lets the decoder process groups of binary error components together.
Joint decoding and degeneracy
Joint sum-product BP (SPA) uses the correlation between X and Z errors on the depolarizing channel. The Komoto–Kasai study reports simulated performance near the hashing-bound benchmark. Later work develops post-processing that uses quantum degeneracy: a correction can succeed when its difference from the actual error is a stabilizer. These results concern the stated depolarizing-channel simulations; circuit-level performance requires a separate evaluation.
The construction and decoder must be considered together. A larger field increases message size and decoding cost, while binary expansion can increase the number of qubits involved in a check. Girth at the finite-field level and girth after binary expansion are also different quantities.
Papers and their roles
- Quantum error correction near the coding theoretical bound — Komoto and Kasai, npj Quantum Information 11, 154 (2025). Construction and joint SPA decoding.
- Quantum Error Correction Exploiting Degeneracy to Approach the Hashing Bound — degeneracy-aware post-processing.
- Quantum Error Correction with Girth-16 Non-Binary LDPC Codes via Affine Permutation Construction — short-cycle control and error-floor evaluation.
- Efficient Mitigation of Error Floors in Quantum Error Correction using Non-Binary Low-Density Parity-Check Codes — cycle-specific decoding methods.
A related development in PP codes
The pair-partition (PP) construction provides another starting point for finite-field assignment. Its quaternary extension constructs new qubit codes together with low-weight, complete canonical logical bases.