Translation-invariant self-correcting stabilizer memory in three dimensions
In plain words
The known three-dimensional candidates either fail eventually at large sizes or give up the regular, repeating structure of a crystal. Whether a perfectly periodic three-dimensional system can be a true passive quantum memory is open.
Precise statement
Does a translation-invariant local commuting Pauli stabilizer Hamiltonian on a 3D cubic lattice exist whose memory time at fixed $T > 0$ grows without bound in system size $L$? Haah's cubic code has an energy barrier $\sim \log L$, giving memory time growing with $L$ only up to $L \sim \exp(c/T)$; Yoshida (2011) excludes stabilizer codes with both translation and scale symmetry.
What would settle it
An explicit translation-invariant 3D stabilizer code with a proof of unbounded memory time at fixed T, or a no-go theorem for this class.
Status in the literature
Unverified note
Layer codes (2023-2025) and cored product codes (2026) give partial self-correction; the 2026 full construction breaks translation symmetry.