Homological Product Code

A homological product code is a quantum error correction code constructed by taking a topological product of two chain complexes. The product operation, introduced by Bravyi and Hastings in 2014, amplifies the distance of the component codes while preserving their sparsity — producing codes with better parameters than either parent.

The breakthrough quantum LDPC codes of 2022 use homological products of expander graphs to achieve constant rate and linear distance. This construction transforms the local expansion of the graph into global error-correcting power, making homological products the key ingredient in asymptotically good quantum LDPC codes.