  A big open question in the quantum information theory concerns feasibility of
a self-correcting quantum memory. A quantum state recorded in such memory can
be stored reliably for a macroscopic time without need for active error
correction if the memory is put in contact with a cold enough thermal bath. In
this paper we derive a rigorous lower bound on the memory time $T_{mem}$ of the
3D Cubic Code model which was recently conjectured to have a self-correcting
behavior. Assuming that dynamics of the memory system can be described by a
Markovian master equation of Davies form, we prove that $T_{mem}\ge L^{c\beta}$
for some constant $c>0$, where $L$ is the lattice size and $\beta$ is the
inverse temperature of the bath. However, this bound applies only if the
lattice size does not exceed certain critical value $L^*\sim e^{\beta/3}$. We
also report a numerical Monte Carlo simulation of the studied memory indicating
that our analytic bounds on $T_{mem}$ are tight up to constant coefficients. In
order to model the readout step we introduce a new decoding algorithm which
might be of independent interest. Our decoder can be implemented efficiently
for any topological stabilizer code and has a constant error threshold under
random uncorrelated errors.
