{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"0e22f815af6baec54ff395254dca04cfde180dfe89a12a4a3f00abb6d628f068","created":"2026-10-03T07:18:09Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"4828049179ca30a562a90109c545b9576129e7ce8c0b1c1ab52f5ada35b3ecda","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"yes-no","assisted_by":[],"external_id":"qi.fault-tolerance-noise.long-range-correlated-threshold","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"Error correction is proven to work when errors on distant qubits are nearly independent. It is unknown whether it still works when each qubit feels a fixed amount of noise but a shared environment links the noise on far-apart qubits with correlations that fall off slowly with distance.","posed_since":"2006","precise":"Qubits on a $D$-dimensional lattice couple through $H_{\\mathrm{int}} = \\sum_i \\sigma_i B_i$ to a Gaussian bath with fixed single-qubit noise strength $\\langle B_i B_i\\rangle$ and spatial correlations $\\langle B_i(t) B_j(t')\\rangle$ decaying as $\\left|r_i - r_j\\right|^{-\\alpha}$. A threshold is proved when the system-bath coupling decays faster than $1/r^D$ (Aharonov, Kitaev and Preskill, Phys. Rev. Lett. 96, 050504, 2006, arXiv:quant-ph/0510231). Decide whether a positive threshold in the single-qubit noise strength exists for $\\alpha \\le D$, or prove a logical-error floor independent of code size.","problem_ref":null,"references":"","settled_by":"A threshold proof for the stated Gaussian-bath model with $\\alpha \\le D$, or a proof of a size-independent logical-error floor for it.","status_note":"Rigorous threshold proofs (2006, 2009, 2013) assume summable correlations; the slowly decaying case has only non-rigorous treatments.","title":"Threshold theorem for noise with slowly decaying spatial 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