Homological product code[1][2]

Description

CSS code formulated using the CSS chain-complex construction in the homological product construction. 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

Zoo code information

Internal code ID: homological_product

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: homological_product

Cite as:
“Homological product code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/homological_product
BibTeX:
@incollection{eczoo_homological_product, title={Homological product code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/homological_product} }
Permanent link:
https://errorcorrectionzoo.org/c/homological_product

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”. 1301.1363
[2]
Sergey Bravyi and Matthew B. Hastings, “Homological Product Codes”. 1311.0885
[3]
T. Jochym-O'Connor, “Fault-tolerant gates via homological product codes”, Quantum 3, 120 (2019). DOI; 1807.09783
[4]
N. Delfosse and M. B. Hastings, “Union-Find Decoders For Homological Product Codes”, Quantum 5, 406 (2021). DOI; 2009.14226
[5]
Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell, “Fiber Bundle Codes: Breaking the $N^{1/2} \operatorname{polylog}(N)$ Barrier for Quantum LDPC Codes”. 2009.03921

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/qldpc/homological_product.yml.