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Phase Structure of the Random-Plaquette Z_2 Gauge Model: Accuracy Threshold for a Toric Quantum Memory
We study the phase structure of the random-plaquette Z_2 lattice gauge model
in three dimensions. In this model, the "gauge coupling" for each plaquette is
a quenched random variable that takes the value \beta with the probability 1-p
and -\beta 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 -\beta, 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 \beta, it is known that there exists a second-order
phase transition at a certain critical "temperature", T(\equiv \beta^{-1}) =
T_c =1.31, which separates an ordered(Higgs) phase at T<T_c and a
disordered(confinement) phase at T>T_c. As p increases, the critical
temperature T_c(p) decreases. In the p-T plane, the curve T_c(p) intersects
with the Nishimori line T_{N}(p) at the certain point (p_c, T_{N}(p_c)). The
value p_c 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 T_c(p) numerically. The accuracy threshold is
estimated as p_c \simeq 0.033.Comment: 24 pages, 14 figures, some clarification