Joint symmetry and dynamical accessibility in compact Hamiltonian encodings of set cover

• URL: http://arxiv.org/abs/2608.11503v1
• Date: Tue, 11 Aug 2026 23:39:28 GMT
• Title: Joint symmetry and dynamical accessibility in compact Hamiltonian encodings of set cover
• Abstract summary: We present a compact multi-register encoding of Minimum Set Cover.<n>We distinguish its fixed, symmetry-allowed space from the generally smaller cyclic space generated by the protocol.<n>We make no quantum-speedup claim: the result is a rigorous separation of global, symmetry-allowed, and dynamically accessible spectral structure.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Which part of a Hamiltonian spectrum is physically relevant when both the initial state and interpolation preserve several symmetries? We address this question for a compact multi-register encoding of Minimum Set Cover. The represented symmetry combines register permutations with the faithful base action of the incidence automorphism group. We distinguish its fixed, symmetry-allowed space from the generally smaller cyclic space generated by the protocol and define gaps relative to isolated bands in that space. A sector-resolved Schur-complement bound certifies cover-measurement probability whenever the sector contains feasible covers and has positive invalid-state separation; the sector kinetic floor is essential. Exact orbit quotients expose inter-sector coincidences invisible to a symmetric protocol, while a stability theorem shows that transverse dark crossings persist under small symmetry-preserving perturbations. On an even-cycle family, the original linear interpolation has an exact global multiplicity closure while the joint-fixed excitation gap remains constant. For the same family we construct a Johnson/Metropolis parent path from a Dicke state to a Gibbs-amplitude state with cover probability $1-O(n^{-5})$. A zero-range comparison and a log-concave three-box coupling give the uniform cyclic-gap certificate $Ω(n^{-13})$. Quantitative derivative bounds then yield a polynomial adiabatic runtime in the abstract Hamiltonian-access model, conditional on Dicke-state preparation and access to the parent Hamiltonian. We make no quantum-speedup claim: the result is a rigorous separation of global, symmetry-allowed, and dynamically accessible spectral structure. Changing the initial state or breaking a preserved symmetry changes that accessible spectrum.

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