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 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 \({\mathbb{F}}_q = \mathbb{F}_q\); 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. 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.

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].