## 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 code can be defined by \(LP(A,B)\), where \(A\) and \(B\) are a pair of matrices with elements from a group algebra. Heuristically, the code is constructed as a hypergraph product code over the group algebra, with each entry subsequently extended into a matrix.

More technically, 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 group-algebraic version of the lifting procedure of protograph LDPC codes. 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

## Parent

- Balanced product (BP) code — Coarsely speaking, a lifted product is a balanced product where the group \(G\) acts freely. However, in principle, a lifted product can be defined for rings that are more general than group algebras \( \mathbb{F}_q G \).

## Children

- Dinur-Hsieh-Lin-Vidick (DHLV) code — DHLV codes are LP codes [5; Footnote 7].
- Hypergraph product (HGP) code — Lifted-product codes for trivial lift are hypergraph-product codes.
- Lift-connected surface (LCS) code
- Toric code — A lifted-product code for the ring \(R=\mathbb{F}_2[x,y]/(x^L-1,y^L-1)\) is the toric code [2; Appx. B].
- Abelian LP code
- Expander LP code
- Two-block group-algebra (2BGA) codes — 2BGA codes are LP\((a,b)\) codes, constructed from a pair of one-by-one matrices \(a,b\in \mathbb{F}_q[G]\) in a group algebra.

## Cousins

- Fiber-bundle code — Lifted products of a length-one with a length-\(m\) chain complex can be thought of as fiber-bundle codes.
- Haah cubic code (CC) — A lifted-product code constructed with coefficients in the ring \(R=\mathbb{F}_2[x,y,z]/(x^L-1,y^L-1,z^L-1)\) is a cubic code [2; Appx. B].
- Fiber-bundle code — The specific fiber-bundle QLDPC code achieving a distance scaling better than \(\sqrt{n}~\text{polylog}(n)\) can also be formulated as an LP code (see published version [6]).
- Subsystem lifted-product (SLP) code — SLP codes reduce to (subspace) LP codes when there is no gauge subsystem.
- Two-block CSS code — LP codes can be constructed using non-square matrices and taking a hypergraph product over a group algebra, while two-block CSS codes are constructed directly using square matrices.

## References

- [1]
- P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021) arXiv:1904.02703 DOI
- [2]
- P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
- [3]
- P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022) arXiv:2012.04068 DOI
- [4]
- N. P. Breuckmann and J. N. Eberhardt, “Quantum Low-Density Parity-Check Codes”, PRX Quantum 2, (2021) DOI
- [5]
- P. Panteleev and G. Kalachev, “Maximally Extendable Sheaf Codes”, (2024) arXiv:2403.03651
- [6]
- M. B. Hastings, J. Haah, and R. O’Donnell, “Fiber bundle codes: breaking the n \({}^{\text{1/2}}\) polylog( n ) barrier for Quantum LDPC codes”, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (2021) DOI

## Page edit log

- Victor V. Albert (2022-01-17) — most recent
- Finnegan Voichick (2021-12-14)
- Pavel Panteleev (2021-11-30)

## Cite as:

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