Balanced Product Quantum Codes

This work provides the first explicit and non-random family of [[N,K,D]] LDPC quantum codes which encode K ∈ Θ(N
^4/5
) logical qubits with distance D ∈ Ω(N
^3/5
). The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product. Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to show that there exist families of LDPC quantum codes which break the polylog(N)√N distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally. Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have K ∈ Θ(N) and that we conjecture to have linear distance D ∈ Θ(N).