Dual correspondence between classical spin models and quantum CSS states

The correspondence between classical spin models and quantum states has attracted much attention in recent years. However, it remains an open problem as to which specific spin model a given (well-known) quantum state maps to. In this Letter, we provide such an explicit correspondence for an important class of quantum states where a duality relation is proved between classical spin models and quantum Calderbank-Shor-Steane (CSS) states. In particular, we employ graph-theoretic methods to prove that the partition function of a classical spin model on a hypergraph $H$ is equal to the inner product of a product state with a quantum CSS state on a dual hypergraph $\tilde{H}$. We next use this dual correspondence to prove that the critical behavior of the classical system corresponds to a relative stability of the corresponding CSS state to bit-flip (and phase-flip) noise, thus called critical stability. We finally conjecture that such critical stability is related to the topological order in quantum CSS states, thus providing a possible practical characterization of such states.