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Abstract In the emerging quantum internet, complex network topology could lead to efficient quantum communication and robustness against failures. However, there are concerns about complexity in quantum communication networks, such as potentially limited end-to-end transmission capacity. These challenges call for model systems in which the impact of complex topology on quantum communication protocols can be explored. Here, we present a theoretical model for complex quantum communication networks on a lattice of spins, wherein entangled spin clusters in interacting quantum spin systems serve as communication links between appropriately selected regions of spins. Specifically, we show that ground state Greenberger-Horne-Zeilinger clusters of the two-dimensional random transverse-field Ising model can be used as communication links between regions of spins. Further, the resulting quantum networks can have complexity comparable to that of the classical internet. Our work provides a generative model for further studies towards determining the network characteristics of the emerging quantum internet. 

Realistic modeling of ecological population dynamics requires spatially explicit descriptions that can take into account spatial heterogeneity as well as long-distance dispersal. Here, we present Monte Carlo simulations and numerical renormalization group results for the paradigmatic model, the contact process, in the combined presence of these factors in both one and two-dimensional systems. Our results confirm our analytic arguments stating that the density vanishes smoothly at the extinction threshold, in a way characteristic of infinite-order transitions. This extremely smooth vanishing of the global density entails an enhanced exposure of the population to extinction events. At the same time, a reverse order parameter, the local persistence displays a discontinuity characteristic of mixed-order transitions, as it approaches a non-universal critical value algebraically with an exponent\beta_p'<1 ${\beta }_p\prime <1$. 

In cluster tomography, we propose measuring the number of clusters 𝑁 intersected by a line segment of length ℓ across a finite sample. As expected, the leading order of 𝑁⁡(ℓ) scales as 𝑎⁢ℓ, where 𝑎 depends on microscopic details of the system. However, at criticality, there is often an additional nonlinearity of the form 𝑏⁢ln⁡(ℓ), originating from the endpoints of the line segment. By performing large scale Monte Carlo simulations of both two- and three-dimensional percolation, we find that 𝑏 is universal and depends only on the angles encountered at the endpoints of the line segment intersecting the sample. Our findings are further supported by analytic arguments in two dimensions, building on results in conformal field theory. Being broadly applicable, cluster tomography can be an efficient tool for detecting phase transitions and characterizing the corresponding universality class in classical or quantum systems with a relevant cluster structure.