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 and that 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 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 $s\ge 4$ (which corresponds to 16 logical qubits) and the distance $d\ge 25$, 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 quantum low-density parity-check surface architectures can offer a scalable and resource-efficient path toward near-term fault-tolerant quantum computation.

14 pages, 10 figures; published version

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