Quantum LDPC Codes

• Quantum LDPC codes are quantum stabilizer codes defined by sparse, constant-weight parity-check matrices that facilitate efficient error correction.
• They promise constant overhead and scalability beyond traditional topological codes by employing sophisticated algebraic, combinatorial, and homological techniques.
• Their design addresses the challenge of balancing commutativity, sparsity, and large minimum distance to enhance quantum error correction capabilities.

Quantum low-density parity-check (LDPC) codes are quantum stabilizer codes characterized by sparse, constant-weight parity-check matrices—each stabilizer generator involves only O(1) qubits and each qubit appears in O(1) stabilizers. The advent of quantum LDPC codes marks a milestone in quantum error correction, promising the potential for constant overhead, efficient decoding, and scalability well beyond topological codes such as the surface code. Modern constructions of quantum LDPC codes rely on sophisticated algebraic, combinatorial, and homological techniques to reconcile the conflicting requirements of commutativity, sparsity, and large minimum distance.

1. Fundamental Definitions and Constraints

Quantum LDPC codes are instances of stabilizer codes defined over $n$ qubits by an abelian subgroup $\mathcal S \subset \mathcal P_n$, where the code space $\