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arXiv:hep-th/0310279 · Nuclear Physics B

Self-dual random-plaquette gauge model and the quantum toric code

Koujin Takeda, Hidetoshi Nishimori

Published Mar 25, 2004• 42 citations
arXivHTMLPDF
DOI
10.1016/j.nuclphysb.2004.03.006

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Journal of the Physical Society of Japan•Jan 1, 2005•3 citations
We investigate how the lattice geometry influences the error thresholds of two-dimensional toric codes. In the presence of bit- or phase-flip errors, the toric code maps onto the two-dimensional random-bond Ising model (RBIM). We determine the critical behaviour of the RBIM using replica-exchange Monte Carlo simulations. While previous studies have explored thresholds for general lattice geometries under suboptimal decoders such as minimum-weight perfect matching, the rapid development of near-optimal decoders makes resolving the ultimate, maximum-likelihood code capacity relevant. We compute these optimal error thresholds on the Nishimori line for the square, honeycomb, triangular, dice, and kagome lattices. Owing to lattice duality, the phase-flip threshold on a given lattice is equivalent to the bit-flip threshold on its dual. We find that the optimal thresholds of dual-lattice pairs display a characteristic duality-driven splitting around the self-dual square-lattice, mirroring the qualitative behaviour observed for suboptimal decoders. While the average coordination number has the strongest impact on the error threshold, our results demonstrate that the detailed arrangement of vertices and plaquettes also plays a significant role in determining its precise value.
A characteristic feature of the 3 d plaquette Ising model is its planar subsystem symmetry. The quantum version of this model has been shown to be related via a duality to the X-Cube model, which has been paradigmatic in the new and rapidly developing field of fractons. The relation between the 3 d plaquette Ising and the X-Cube model is similar to that between the 2 d quantum transverse spin Ising model and the Toric Code. Gauging the global symmetry in the case of the 2 d Ising model and considering the gauge invariant sector of the high temperature phase leads to the Toric Code, whereas gauging the subsystem symmetry of the 3 d quantum transverse spin plaquette Ising model leads to the X-Cube model. A non-standard dual formulation of the 3 d plaquette Ising model which utilises three flavours of spins has recently been discussed in the context of dualising the fracton-free sector of the X-Cube model. In this paper we investigate the classical spin version of this non-standard dual Hamiltonian and discuss its properties in relation to the more familiar Ashkin–Teller-like dual and further related dual formulations involving both link and vertex spins and non-Ising spins.
We study the phase diagrams of random models with bimodal randomness, especially the random bond Ising model and the random plaquette gauge model from the viewpoint of quantum information theory. Using Fourier transformation in conjunction with the replica trick, we show that these models have a common structure under duality transformation. This observation enables us to derive a conjecture on the exact location of the multicritical point and lead the accuracy threshold of the toric code, which is one of the topological quantum memories.