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
- Lifted-product (LP) code — Lifted-product codes for trivial group \(G\) are hypergraph-product codes.
- Homological product code — A homological product of chain complexes corresponding to two classical codes is a hypergraph product code [5].
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
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