Threshold theorem for noise with slowly decaying spatial correlations
In plain words
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 statement
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.
What would settle it
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 in the literature
Rigorous threshold proofs (2006, 2009, 2013) assume summable correlations; the slowly decaying case has only non-rigorous treatments.