Finite Rate QLDPC-GKP Coding Scheme that Surpasses the CSS Hamming Bound

Quantum error correction has recently been shown to benefit greatly from specific physical encodings of the code qubits. Many authors have considered the continuous-variable Gottesman-Kitaev-Preskill (GKP) encoding, and then imposed an outer discrete-variable surface code on these GKP qubits. Under such concatenation, the analog information from the inner GKP error correction improves the noise threshold of the outer code. However, the surface code has vanishing rate and is resource intensive. We concatenate the GKP code with generic quantum low-density parity-check (QLDPC) codes and demonstrate a natural way to exploit the GKP analog information (GKP-AI) in iterative decoders. We first show the noise thresholds for two QLDPC code families, and then show the improvements when the hardware-friendly min-sum decoder utilizes the GKP-AI. When the GKP-AI is combined with a sequential update schedule for min-sum, the scheme surpasses the CSS Hamming bound for these code families. Furthermore, we observe that the GKP-AI helps the decoder in escaping harmful trapping sets in the Tanner graph of the QLDPC code, thereby eliminating or significantly lowering the error floor of the logical error rate curves.