## Description

CSS code formulated using the homological product of two chain complexes (see CSS-to-homology correspondence). Stub.

## Protection

Given two codes \([[n_a, k_a, d_a, w_a]]\) for \(a\in\{1,2\}\), where \(w_a\) denotes the maximum hamming weight of all rows and columns of \(\partial_a\), the homological product code has parameter \([[n=n_1 n_2, k=k_1 k_2, d\leq d_1 d_2, w\leq w_1+w_2]]\). From this formula, and the fact that a randomly selected boundary operator \(\partial\) yields a CSS code that is good with high probability, we see that the product code has \(k=\Theta(n)\) and \(w=O(\sqrt{n})\) with high probability. The main result in [2] is to show that the product code actually has linear distance with high probability as well. To sum up, it is shown that we have a family of \([[n,k=c_1 n, d=c_2 n, w=c_3 \sqrt{n}]]\) codes given small enough \(c_1,c_2,c_3\).

## Gates

Universal set of gates can be obtained by fault-tolerantly mapping between different encoded representations of a given logical state [3].

## Decoding

Union-find [4].

## Fault Tolerance

Universal set of gates can be obtained by fault-tolerantly mapping between different encoded representations of a given logical state [3].

## Parent

- Fiber-bundle code — Fiber-bundle code can be viewed as a homological product code with a twisted product.

## Children

- Distance-balanced code — Distance balancing relies on taking homological product of chain complexes corresponding to a classical and a quantum code.
- Hypergraph product code — A homological product of chain complexes corresponding to two classical codes is a hypergraph product code [5].

## Cousin

- Random quantum code — Random homological codes are asymptotically good with high probability [1; Thm. 1].

## References

- [1]
- M. H. Freedman and M. B. Hastings, “Quantum Systems on Non-\(k\)-Hyperfinite Complexes: A Generalization of Classical Statistical Mechanics on Expander Graphs”, (2013) arXiv:1301.1363
- [2]
- S. Bravyi and M. B. Hastings, “Homological Product Codes”, (2013) arXiv:1311.0885
- [3]
- T. Jochym-O’Connor, “Fault-tolerant gates via homological product codes”, Quantum 3, 120 (2019) arXiv:1807.09783 DOI
- [4]
- N. Delfosse and M. B. Hastings, “Union-Find Decoders For Homological Product Codes”, Quantum 5, 406 (2021) arXiv:2009.14226 DOI
- [5]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber Bundle Codes: Breaking the \(N^{1/2} \operatorname{polylog}(N)\) Barrier for Quantum LDPC Codes”, (2020) arXiv:2009.03921

## Page edit log

- Victor V. Albert (2022-11-22) — most recent
- Victor V. Albert (2022-03-14)
- Xinyuan Zheng (2021-12-15)
- Victor V. Albert (2021-12-03)

## Cite as:

“Homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/homological_product