Universal coarse geometry of spin systems

Abstract 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 $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> on an (abstract) spin system with an infinite collection of sites X , we define a universal coarse structure $$\mathcal {E}_{\phi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> </mml:math> on the set X with the property that a state has decay of correlations with respect to a coarse structure $$\mathcal {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> on X if and only if $$\mathcal {E}_{\phi }\subseteq \mathcal {E}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mo>⊆</mml:mo> <mml:mi>E</mml:mi> </mml:mrow> </mml:math> . We show that under mild assumptions, the coarsely connected completion $$(\mathcal {E}_{\phi })_{con}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>ϕ</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>con</mml:mi> </mml:mrow> </mml:msub> </mml:math> is stable under quasi-local perturbations of the state $$\phi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϕ</mml:mi> </mml:math> . 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 $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and the coarse structure of the state $$\psi \circ \alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ψ</mml:mi> <mml:mo>∘</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:math> where $$\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> is any product state.