Lifted-product (LP) code[1][2]


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.


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


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


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




  • 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

Internal code ID: lifted_product

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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.
@incollection{eczoo_lifted_product, title={Lifted-product (LP) code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
Permanent link:


P. Panteleev and G. Kalachev, “Degenerate Quantum LDPC Codes With Good Finite Length Performance”, Quantum 5, 585 (2021). DOI; 1904.02703
Pavel Panteleev and Gleb Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”. 2111.03654
P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022). DOI; 2012.04068
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.