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Universal coarse geometry of spin systems
Quantum Physics · Submitted 12 Nov 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 to a quantum circuit $\alpha$ and the coarse structure of the state $\psi\circ \alpha$ where $\psi$ is any product state.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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? | background 0.96 | 0.05 | 0.03 | unclear 0.74 |
| s1 | 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. | contribution 1.00 | 0.87 | 0.57 | asserted 0.39 |
| s2 | 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 $\mathcal{E}_{\phi}\subseteq \mathcal{E}$. | contribution 0.81 | 0.91 | 0.26 | asserted 1.00 |
| s3 | We show that under mild assumptions, the coarsely connected completion $(\mathcal{E}_{\phi})_{con}$ is stable under quasi-local perturbations of the state $\phi$. | result 0.99 | 0.96 | 0.15 | asserted 1.00 |
| s4 | We also develop in parallel a dynamical coarse structure for arbitrary quantum channels, and prove a similar stability result. | contribution 0.95 | 0.96 | 0.08 | asserted 1.00 |
| s5 | 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 to a quantum circuit $\alpha$ and the coarse structure of the state $\psi\circ \alpha$ where $\psi$ is any product state. | result 0.96 | 0.95 | 0.14 | asserted 1.00 |
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Original 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 $\phi$ 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 to a quantum circuit $\alpha$ and the coarse structure of the state $\psi\circ \alpha$ where $\psi$ is any product state.
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A typical choice is the $n$-dimensional lattice $S=^n$, but it is an interesting question to which extent the discrete geometry of $S$ influences the main features of the model, see e.g. [ref] for existing results in this direction.
$C^\ast$-categorical prefactorization algebras for superselection sectors and topological order · Introduction and summary · 0.80
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