Nonlocal resources for error correction in quantum low-density parity-check codes

Quantum low-density parity-check (qLDPC) codes with high-performance and nonlocal stabilizers are an attractive alternative to the surface code because of their relatively high code rate and distance. However, unlike the surface code that has simple, geometrically local, stabilizer checks, qLDPC codes with nonlocal stabilizers are challenging to measure. Recent advances have shown how to deterministically perform high-fidelity, cavity-mediated many-body gates, enabling the encoding and decoding of nonlocal GHZ states. We show how to use this resource for constant-depth fault-tolerant stabilizer measurements in hypergraph product and lifted product codes. Using circuit-level noise simulations, including leakage and collective error from the noise-optimized cavity-mediated gate, we find promising thresholds of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>0.84</a:mn><a:mo>%</a:mo><a:mo>–</a:mo><a:mn>0.60</a:mn><a:mo>%</a:mo></a:mrow></a:math> for the hypergraph product code and pseudothreshold of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>0.3</b:mn><b:mo>%</b:mo><b:mo>–</b:mo><b:mn>0.4</b:mn><b:mo>%</b:mo></b:mrow></b:math> for the lifted product codes, with cavity cooperativities in the range <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>C</c:mi><c:mo>∼</c:mo><c:msup><c:mn>10</c:mn><c:mn>4</c:mn></c:msup><c:mo>–</c:mo><c:msup><c:mn>10</c:mn><c:mn>6</c:mn></c:msup></c:mrow></c:math>. We propose a trilayer architecture, compatible with neutral atom platforms, which enhances circuit parallelizability.