aiqbench analysis
This paper establishes a rigorous mathematical framework linking quantum automorphism groups to classical graph theory by proving that the natural quantum analogue of compactness reduces to its classical counterpart. By constructing an intermediate convex set between standard and fractional automorphisms, it clarifies where quantum symmetries can strictly refine existing combinatorial theories without introducing new physical phenomena. The work advances theoretical understanding at the intersection of operator algebras and discrete mathematics rather than addressing near-term hardware or algorithmic applications.
Compact graphs and quantum automorphisms
arXiv:2606.13928v1 Announce Type: new Abstract: Compact graphs are graphs for which the fractional automorphism polytope has no genuinely fractional vertices. This paper proposes a quantum analogue of this idea by evaluating the fundamental magic unitary of the quantum automorphism group on states, which we show to produce a closed convex set of doubly stochastic matrices sitting between the classical automorphism polytope and the full fractional automorphism polytope. Our main result is that the natural quantum analogue of compactness is classical, that is, a quantum compact graph is classically compact. We also relate this set to the quantum orbital algebra and obtain a hierarchy of classical and quantum compactness pseudo notions. The framework recovers familiar consequences of compactness through commutants and suggests quantum analogues of generous transitivity and distance-transitivity. We also isolate examples and open problems indicating where quantum symmetries may strictly refine the classical compactness theory.
Abstract:Compact graphs are graphs for which the fractional automorphism polytope has no genuinely fractional vertices. This paper proposes a quantum analogue of this idea by evaluating the fundamental magic unitary of the quantum automorphism group on states, which we show to produce a closed convex set of doubly stochastic matrices sitting between the classical automorphism polytope and the full fractional automorphism polytope. Our main result is that the natural quantum analogue of compactness is classical, that is, a quantum compact graph is classically compact. We also relate this set to the quantum orbital algebra and obtain a hierarchy of classical and quantum compactness pseudo notions. The framework recovers familiar consequences of compactness through commutants and suggests quantum analogues of generous transitivity and distance-transitivity. We also isolate examples and open problems indicating where quantum symmetries may strictly refine the classical compactness theory.
Quantum Physics (quant-ph); Combinatorics (math.CO)
arXiv:2606.13928 [quant-ph]
(or arXiv:2606.13928v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.13928 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 237 words
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