A strong-weak coupling duality between two perturbed quantum many-body systems: CSS codes and Ising-like systems

摘要

Graphs and recently hypergraphs have been known as an important tool forconsidering different properties of quantum many-body systems. In this paper,we study a mapping between an important class of quantum systems namely quantumCalderbank-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 aIsing-like system on dual hypergraph $\tilde{H}$ in a transverse field.Interestingly, we show that a strong regime of couplings in one of the systemsis mapped to a weak regime of couplings in another one. We also give someapplications for such a mapping where we study robustness of differenttopological CSS codes against a uniform magnetic field including Kitaev's toriccodes defined on graphs and color codes in different dimensions. We show that aperturbed Kitaev's toric code on an arbitrary graph is mapped to an Ising modelin a transverse field on the same graph and a perturbed color code on a D-colexis mapped to a Ising-like model on a D-simplicial lattice in a transversefield. In particular, we use these results to explicitly compare the robustnessof TC on different graphs in different dimensions. Interestingly, our resultsshow that the robustness of such topological codes defined on graphs decreaseswith increasing dimension. Furthermore, we also use the duality mapping forsome self-dual models where we exactly derive the point of phase transition.