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Phase Structure of the Random-Plaquette Z_2 Gauge Model: Accuracy Threshold for a Toric Quantum Memory

Takuya Ohno·Gaku Arakawa·Ikuo Ichinose·Tetsuo Matsui
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Abstract

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 βwith the probability 1-p and -βwith 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(\equiv β^{-1}) = T_c =1.31, which separates an ordered(Higgs) 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.

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Takuya Ohno, Gaku Arakawa, Ikuo Ichinose, Tetsuo Matsui. 2004-05-07. Phase Structure of the Random-Plaquette Z_2 Gauge Model: Accuracy Threshold for a Toric Quantum Memory. https://doi.org/10.1016/j.nuclphysb.2004.07.003

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