Coherent error threshold for quantum LDPC codes

This paper establishes the first general rigorous coherent-error threshold for quantum low-density parity-check codes. For any qLDPC family whose distance satisfies d=Ω(log n), arbitrary local coherent noise from a constant-depth circuit, including non-Pauli Hermitian generators acting on at most r qubits, can be suppressed below a nonzero noise strength: the logical error in diamond distance decays exponentially with code distance, even when errors interfere within the same syndrome sector. The result holds for both optimal recovery and the practical minimum-weight Pauli decoder, and extends from unitary coherent noise to local CPTP channels. The central innovation is cluster resummation: instead of taking norms configuration by configuration, the proof isolates a connected error cluster and exactly resums all disconnected errors first, avoiding exponential dependence on system size. A local channel term has diamond norm at most 2 sin|η|, and convergence follows when the large-cluster activity satisfies Ξ<1; the resulting optimal-recovery bound scales as 2√[D n/(χ(1−q))]q^{d/(2r)}, with q=2χ sin|η|. Minimum-weight decoding has a related exponential bound with a larger polynomial prefactor and a lower guaranteed threshold. The theorem covers topological, expander, bivariate bicycle, and asymptotically good qLDPC families, broadening rigorous guarantees beyond linear-distance constructions. It complements experimental scaling efforts such as Google Quantum AI while identifying noisy syndrome extraction and non-Markovian noise as the next major gaps.