Confinement-Higgs transition in a disordered gauge theory and the accuracy threshold for quantum memory

Abstract

We study the ±J random-plaquette Z2 gauge model (RPGM) in three spatial dimensions, a three-dimensional analog of the two-dimensional ±J random-bond Ising model (RBIM). The model is a pure Z2 gauge theory in which randomly chosen plaquettes (occuring with concentration p) have couplings with the "wrong sign" so that magnetic flux is energetically favored on these plaquettes. Excitations of the model are one-dimensional "flux tubes" that terminate at "magnetic monopoles" located inside lattice cubes that contain an odd number of wrong-sign plaquettes. Electric confinement can be driven by thermal fluctuations of the flux tubes, by the quenched background of magnetic monopoles, or by a combination of the two. Like the RBIM, the RPGM has enhanced symmetry along a "Nishimori line" in the p-T plane (where T is the temperature). The critical concentration pc of wrong-sign plaquettes at the confinement-Higgs phase transition along the Nishimori line can be identified with the accuracy threshold for robust storage of quantum information using topological error-correcting codes: if qubit phase errors, qubit bit-flip errors, and errors in the measurement of local check operators all occur at rates below pc, then encoded quantum information can be protected perfectly from damage in the limit of a large code block. Through Monte-Carlo simulations, we measure pc0, the critical concentration along the T = 0 axis (a lower bound on pc), finding pc0 = .0293 ± .0002. We also measure the critical concentration of antiferromagnetic bonds in the two-dimensional RBIM on the T = 0 axis, finding pc0 = .1031 ± .0001. Our value of pc0 is incompatible with the value of pc = .1093 ± .0002 found in earlier numerical studies of the RBIM, in disagreement with the conjecture that the phase boundary of the RBIM is vertical (parallel to the T axis) below the Nishimori line. The model can be generalized to a rank-r antisymmetric tensor field in d dimensions, in the presence of quenched disorder.

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

“…After that many interesting studies on Kitaev's model appeared [2,3,4,5,6,7,8]. Among them, two of the present authors [7] generalized Kitaev's model and also clarified the relationship between Kitaev's model and a spontaneously broken U (1) gauge theory.…”

“…[4], the critical value of p along the T = 0 axis was estimated as 0.029, which corresponds p 0 in Fig.1. This value gives a lower bound of p c , and Fig.9 shows that our MC simulations are consistent with their calculations.…”

“…[3,4], p c of the toric code can be obtained by studying the RBIM in 2D and/or the Z 2 RPGM in 3D. We note that p c depends on the practical methods of error corrections which one employs.…”

“…In the method of Ref. [4] it is estimated as p c = 0.029. Here we shall review the estimation of p c by using the statistical-mechanical models in two steps.…”

“…As discussed in Ref. [4], it is obtained for Kitaev's model by studying the random-plaquette gauge model(RPGM) on a three-dimensional (3D) lattice with Z 2 gauge symmetry. In the RPGM, the (inverse) gauge coupling for each plaquette takes the values ±β with random sign; the probability to take β is given by 1 − p and the probability of "wrong-sign" −β is p. The original nonrandom (p = 0) 3D Z 2 lattice gauge theory is known to be dual to the classical 3D Ising model.…”