Lifted-product (LP) code [1–3]

Description

Galois-qudit 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 a hypergraph product over the group algebra \(\mathbb{F}_q[G]\), with each entry then replaced by a \(|G|\times|G|\) matrix. The code thus has \(|G|\) times as many qudits and checks as a hypergraph product of matrices of the same shapes [2; Sec. III]. For \(q=2\), each block’s row and column weight equals the number of group elements in its entry [2; Sec. III]. Entries that are single group elements therefore keep the check weights of a hypergraph product with ones in their place.

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 \(\mathbb{F}_q\); 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 first version of this construction contains hypergraph-product codes in the case where one of the two input parity-check matrices is square [1].

The key operation behind the \(G\)-lifted product is the \(G\)-lift, a group-algebraic version of the lifting procedure of protograph LDPC codes. The lifted product is obtained from the usual hypergraph product of the two lifted complexes by quotienting by the shared \(G\) action [3].

Lifted products over non-Abelian group algebras have been studied, e.g., over the dihedral group algebra \(\mathbb{F}_q[D_{2n}]\), whose Wedderburn decomposition yields short non-Abelian quantum moderate-density parity-check codes—dubbed dihedral quantum codes—with an explicit dimension formula and distance bound [4]. Non-Abelian lifts also underlie mitten codes, whose non-commutativity removes a distance ceiling that constrains Abelian LP codes of the same base-matrix shape [5].