Threshold theorem for noise with slowly decaying spatial correlations · revision history

Revision 2

Accepted · core change · record

Changes against revision 1:

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 · changed · core
Revision 1: 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.
Revision 2: Qubits on a $D$-dimensional lattice couple through $H_{\int} = \sum_i \sigma_i B_i$ 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

Decisions