Balanced Product Quantum Codes

TL;DR

This paper introduces the first explicit family of LDPC quantum codes with polynomially large distance and logical qubits, constructed via balanced products of classical codes and Ramanujan graphs, unconditionally surpassing previous probabilistic bounds.

Contribution

It presents the first explicit, non-random LDPC quantum codes with polynomially large parameters using balanced product construction, including non-abelian twists.

Findings

Codes encode $K o heta(N^{4/5})$ qubits.

Distance $D o ext{Omega}(N^{3/5})$.

Potential for linear distance and qubits with further development.

Abstract

This work provides the first explicit and non-random family of $[[N,K,D]]$ LDPC quantum codes which encode $K \in \Theta(N^\frac{4}{5})$ logical qubits with distance $D \in \Omega(N^\frac{3}{5})$. 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 $\operatorname{polylog}(N)\sqrt{N}$ 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 $K\in \Theta(N)$ and that we conjecture to have linear distance $D\in \Theta(N)$.