Universal coarse geometry of spin systems

The prospect of realizing highly entangled states on quantum processors with fundamentally different hardware geometries raises the question: to what extent does a state of a quantum spin system have an intrinsic geometry? In this paper, we propose that both states and dynamics of a spin system have a canonically associated coarse geometry, in the sense of Roe, on the set of sites in the thermodynamic limit. For a state \(\phi \) on an (abstract) spin system with an infinite collection of sites X, we define a universal coarse structure \(\mathcal {E}_{\phi }\) on the set X with the property that a state has decay of correlations with respect to a coarse structure \(\mathcal {E}\) on X if and only if \(\mathcal {E}_{\phi }\subseteq \mathcal {E}\). We show that under mild assumptions, the coarsely connected completion \((\mathcal {E}_{\phi })_{con}\) is stable under quasi-local perturbations of the state \(\phi \). We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. We show that several order parameters of a state only depend on the coarse structure of an underlying spatial metric, and we establish a basic compatibility between the dynamical coarse structure associated with a quantum circuit \(\alpha \) and the coarse structure of the state \(\psi \circ \alpha \) where \(\psi \) is any product state.