  The ability to store information is of fundamental importance to any
computer, be it classical or quantum. To identify systems for quantum memories
which rely, analogously to classical memories, on passive error protection
(`self-correction') is of greatest interest in quantum information science.
While systems with topological ground states have been considered to be
promising candidates, a large class of them was recently proven unstable
against thermal fluctuations. Here, we propose two-dimensional (2D) spin models
unaffected by this result. Specifically, we introduce repulsive long-range
interactions in the toric code and establish a memory lifetime polynomially
increasing with the system size. This remarkable stability is shown to
originate directly from the repulsive long-range nature of the interactions. We
study the time dynamics of the quantum memory in terms of diffusing anyons and
support our analytical results with extensive numerical simulations. Our
findings demonstrate that self-correcting quantum memories can exist in 2D at
finite temperatures.
