## Description

Code that utilizes the notion of a lifted product in its construction. Lifted products of certain classical Tanner codes are the first (asymptotically) good QLDPC codes.

A lifted product over a ring \(R\) is a product of two chain complexes whose chains are free modules over \(R\). An interesting case is when \(R=\mathbb{F}_q G\), the group-\(G\) algebra over the finite field \({\mathbb{F}}_q = GF(q)\); in this case, the product can be called a \(G\)-lifted product. Just like its further generalization the balanced product, a lifted product code generalizes a hypergraph product code in that a reduction of symmetry is exploited to decrease the number of physical qubits required.

The key operation behind the \(G\)-lifted product is the \(G\)-lift. A \(G\)-lift of a \(\mathbb{F}_q\)-valued matrix \(A\) substitutes matrix elements of \(A\) with matrices forming the regular representation of the group algebra \({\mathbb{F}}_q G\) according to some rule. A combination of the lift and the usual hypergraph product yields lifted-product codes. The two operations commute: one can first take the usual hypergraph product of two chain complexes, and then lift the resulting product complex; equivalently, one can take the hypergraph product of the two lifted complexes.

## Protection

## Rate

## Notes

## Parents

## Children

- Expander lifted-product code — Lifted products of certain classical Tanner codes are the first asymptotically good QLDPC codes.
- Hypergraph product code — Lifted-product codes for trivial group \(G\) are hypergraph-product codes.

## Cousins

- Kitaev surface code — A lifted product code for the ring \(R=\mathbb{F}_2[x,y]/(x^L-1,y^L-1)\) is the toric code.
- Haah cubic code — A lifted product code for the ring \(R=\mathbb{F}_2[x,y,z]/(x^L-1,y^L-1,z^L-1)\) is the cubic code.

## Zoo code information

## References

- [1]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021). DOI; 1904.02703
- [2]
- Pavel Panteleev and Gleb Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”. 2111.03654
- [3]
- P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022). DOI; 2012.04068
- [4]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021). DOI; 2103.06309

## Cite as:

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