Square homological product code

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 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].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Fiber-bundle codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Homological product code

Parents

Homological product code

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 (2025-03-03) — most recent

Cite as:

“Square homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/square_homological_product

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/qldpc/homological/balanced_product/square_homological_product.yml.