Square homological product code[1]
Alternative Names: Single-sector homological code, Bravyi-Hastings homological code, Square tensor product code.
Description
Homological product code whose underlying quantum-code boundary operators are square matrices (see Qubit CSS-to-homology correspondence).
Each base code is associated with the chain complex \( C_i \longrightarrow C_i\longrightarrow C_i\) such that the boundary operator (a.k.a. parity-check matrix) satisfies \(H_i^{2}=0\) [2; Def. 3.8]. The parity-check matrix of the resulting product code is \begin{align} H_1 \otimes I_2 + I_1 \otimes H_2~, \tag*{(1)}\end{align} where \(I_i\) is the identity on the check space of code \(i\). The logical dimension \(k = k_1 k_2\).
Protection
Square homological-product codes admit different properties than those with rectangular boundary operators [2; Sec. 3.4].Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Homological product codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Square homological product codes are homological product codes whose boundary operators are square matrices [2; Sec. 3.4].
Square homological product code
References
- [1]
- S. Bravyi and M. B. Hastings, “Homological Product Codes”, (2013) arXiv:1311.0885
- [2]
- B. Audoux and A. Couvreur, “On tensor products of CSS Codes”, (2018) arXiv:1512.07081
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2025-03-03)
Cite as:
“Square homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/square_homological_product, arXiv:2606.11484