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Title: Threshold theorem for noise with slowly decaying spatial correlations
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.
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.
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.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status note: Rigorous threshold proofs (2006, 2009, 2013) assume summable correlations; the slowly decaying case has only non-rigorous treatments.
Posed since: 2006