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The small-world effect is a universal feature used to explain many differentphenomena like percolation, diffusion, and consensus. Starting from any regularlattice of $N$ sites, the small-world effect can be attained by rewiringrandomly an $\mathcal{O}(N)$ number of links or by superimposing an equivalentnumber of new links onto the system. In a classical system this procedure isknown to change radically its critical point and behavior, the new system beingalways effectively mean-field. Here, we prove that at the quantum level theabove scenario does not apply: when an $\mathcal{O}(N)$ number of new couplingsare randomly superimposed onto a quantum Ising chain, its quantum criticalpoint and behavior both remain unchanged. In other words, at zero temperaturequantum fluctuations destroy any small-world effect. This exact result shedsnew light on the significance of the quantum critical point as athermodynamically stable feature of nature that has no analogous at theclassical level and essentially prevents a naive application of network theoryto quantum systems. The derivation is obtained by combining thequantum-classical mapping with a simple topological argument.

Absence of small-world effects at the quantum level and stability of the quantum critical point