Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code

Abstract

We propose and analyze a hierarchical quantum error correction (QEC) scheme that concatenates hypergraph product (HGP) codes with rotated surface codes, which is compatible with quantum computers with only nearest-neighbor interactions. The outer code employs (3,4)-random HGP codes, known for their constant encoding rate and favorable distance scaling, while the inner code consists of a rotated surface code with distance 5, allowing hardware compatibility through lattice surgery. To address the decoding bottleneck, we utilize a soft-decision decoding strategy that combines belief propagation with ordered statistics (BP-OS) decoding, enhanced by a syndrome-conditioned logical error probability computed via a tailored lookup table for the inner code. Numerical simulations under a code capacity noise model demonstrate that our hierarchical codes achieve logical error suppression below the threshold. Furthermore, we derive explicit conditions under which the proposed codes surpass surface codes as a QEC code in both qubit efficiency and error rate. In particular, for the size parameter s ≥ 4 (which corresponds to 16 logical qubits) and the distance d ≥ 25, our construction outperforms the rotated surface code in practical regimes with physical error rates around or less than 10−2. These results suggest that concatenated qLDPC-surface architectures can offer a scalable and resource-efficient path toward near-term fault-tolerant quantum computation.