Hypergraph product code[1][2]

Description

A family of \([[n,k,d]]\) CSS codes whose construction is based on two binary linear seed codes \(C_1\) and \(C_2\).

Protection

The hypergraph product has distance \(d=O(\sqrt{n})\). The number of encoded logical qubits is \(k=O(k_1k_2)\) where \(k_1\) and \(k_2\) are the dimensions of the classical seed codes \(C_1\) and \(C_2\).

Transversal Gates

Hadamard (up to logical SWAP gates) and control-\(Z\) on all logical qubits [3].

Gates

Code deformation techniques yield Clifford gates [4].

Parents

Child

Cousins

  • Kitaev surface code — Planar (toric) code obtained from hypergraph product of two repetition (cyclic) codes.
  • XYZ product code — The XYZ product code is based on a hypergraph product of three classical codes.

Zoo code information

Internal code ID: hypergraph_product

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: hypergraph_product

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

References

[1]
J.-P. Tillich and G. Zemor, “Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength”, IEEE Transactions on Information Theory 60, 1193 (2014). DOI; 0903.0566
[2]
A. A. Kovalev and L. P. Pryadko, “Improved quantum hypergraph-product LDPC codes”, 2012 IEEE International Symposium on Information Theory Proceedings (2012). DOI; 1202.0928
[3]
Armanda O. Quintavalle, Paul Webster, and Michael Vasmer, “Partitioning qubits in hypergraph product codes to implement logical gates”. 2204.10812
[4]
A. Krishna and D. Poulin, “Fault-Tolerant Gates on Hypergraph Product Codes”, Physical Review X 11, (2021). DOI; 1909.07424
[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:

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

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