Strong-weak coupling duality between two perturbed quantum many-body systems: Calderbank-Shor-Steane codes and Ising-like systems

Abstract

Graphs and recently hypergraphs have been known as an important tool for considering different properties of quantum many-body systems. In this paper, we study a mapping between an important class of quantum systems, namely quantum Calderbank-Shor-Steane (CSS) codes, and Ising-like systems by using hypergraphs. We show that the Hamiltonian corresponding to a CSS code on a hypergraph $H$ which is perturbed by a uniform magnetic field is mapped to Hamiltonian of a Ising-like system on dual hypergraph $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}$ in a transverse field. Interestingly, we show that a strong regime of couplings in one of the systems is mapped to a weak regime of couplings in another one. We also give some applications for such a mapping where we study robustness of different topological CSS codes against a uniform magnetic field including Kitaev's toric codes defined on graphs and color codes in different dimensions. We show that a perturbed Kitaev's toric code on an arbitrary graph is mapped to an Ising model in a transverse field on the same graph and a perturbed color code on a $D$ colex is mapped to a Ising-like model on a $D$-simplicial lattice in a transverse field. In particular, we use these results to explicitly compare the robustness of toric codes to uniform magnetic-field perturbations on different graphs. Interestingly, our results show that the robustness of such topological codes defined on graphs decreases with increasing dimension. Furthermore, we also use the duality mapping for some self-dual models where we exactly derive the point of phase transition.