Lifted-product (LP) code[1,2] 

Also known as Panteleev-Kalachev (PK) code.

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

Code performance strongly depends on the group \(G\) used in the product [3].

Rate

There is no known simple way to compute the logical dimension \(k\) in the general case [3].

Notes

Formerly known as generalized hypergraph product codes [1], and later renamed to lifted-product codes [3,4].

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

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 quantum code — LP codes can be constructed using non-square matrices and taking a hypergraph product over a group algebra, while two-block quantum 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
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Zoo Code ID: lifted_product

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

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/qldpc/lp/lifted_product.yml.