Analytic and numerical demonstration of quantum self-correction in the 3D Cubic Code

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_memge L^cβ for some constant c>0, where L is the lattice size and β 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^β/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.