Non-local resources for error correction in quantum LDPC codes

Quantum low density parity check (qLDPC) codes are an attractive alternative to the surface code due to their relatively high code rate and distance. However, unlike the surface code which has simple, geometrically local, stabilizer checks, high performing qLDPC codes have non-local stabilizers that are challenging to measure. Recent advancements have shown how to deterministically perform high-fidelity, cavity mediated many-body gates, enabling the encoding and decoding of non-local GHZ states. We integrate this non-local resource into the DiVincenzo-Aliferis method of fault-tolerant stabilizer measurement for quantum hypergraph product and lifted product codes. Using circuit-level noise simulations, including the noise optimized cavity mediated gate, we find promising thresholds of $0.84 \%-0.60 \%$ for the hypergraph product code and psuedo-threshold of $0.3\%-0.4\%$ for the lifted product codes, with cavity cooperativities in the range $C\sim 10^4-10^6$. We propose a compatible tri-layer architectural layout for scheduling stabilizer measurements, enhancing circuit parallelizability.