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

Also called a Panteleev-Kalachev (PK) code. 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.