Existence of a self-correcting quantum memory in three dimensions
In plain words
A four-dimensional version of a known quantum code keeps its stored qubit safe from heat for a time that grows exponentially with size, and two-dimensional versions of this code family cannot. Whether some three-dimensional system can do it was open for over twenty years.
Precise statement
Does there exist a family of geometrically local Hamiltonians on qubits in $R^{3}$, with bounded interaction strength and qubit density, whose ground space encodes at least one qubit with a memory time, under a weakly coupled thermal bath (Davies dynamics), that grows as $\operatorname{exp}(c L^{b})$, $b > 0$, at some fixed temperature $T > 0$?
What would settle it
A proof of exponential memory time for an explicit 3D local Hamiltonian, checked independently.
Status in the literature
Unverified note
A May 2026 preprint (Balasubramanian, Davydova and Lin, arXiv:2605.10943) constructs a non-translation-invariant 3D Pauli stabilizer Hamiltonian with provably exponential memory time below a critical temperature; independent verification is pending.