Phase structure of the random-plaquette Z2 gauge model: accuracy threshold for a toric quantum memory

We study the phase structure of the random-plaquette Z2 lattice gauge model in three dimensions. In this model, the 'gauge coupling' for each plaquette is a quenched random variable that takes the value β with the probability 1-p and -β with the probability p. This model is relevant for the recently proposed quantum memory of toric code. The parameter p is the concentration of the plaquettes with 'wrong-sign' couplings -β, and interpreted as the error probability per qubit in quantum code. In the gauge system with p=0, i.e., with the uniform gauge couplings β, it is known that there exists a second-order phase transition at a certain critical 'temperature', T(=β-1)=Tc=1.31, which separates an ordered (Higgs) phase at TTc and a disordered (confinement) phase at T>Tc. As p increases, the critical temperature Tc(p) decreases. In the p-T plane, the curve Tc(p) intersects with the Nishimori line TN(p) at the certain point (pc,TN(pc)). The value pc is just the accuracy threshold for a fault-tolerant quantum memory and associated quantum computations. By the Monte Carlo simulations, we calculate the specific heat and the expectation values of the Wilson loop to obtain the phase-transition line Tc(p) numerically. The accuracy threshold is estimated as pc∼0.033