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This paper investigates the spectral properties of Hamiltonians encoding fixed-cardinality Set Cover problems, specifically analyzing how symmetries and initial states constrain dynamical accessibility in quantum adiabatic algorithms. The authors derive rigorous bounds on measurement probabilities and excitation gaps to demonstrate that global multiplicity can be dynamically suppressed, resulting in a protocol-dependent separation of spectral sectors rather than a claimed quantum speedup.
Orbit-resolved spectra and dynamical accessibility in multi-register covering Hamiltonians
arXiv:2608.11503v2 Announce Type: replace Abstract: Which part of a Hamiltonian spectrum is relevant when the initial state and interpolation preserve multiple symmetries? We study this question for a multi-register encoding of fixed-cardinality Set Cover. Register permutations and incidence automorphisms define a joint action whose orbits are unlabelled candidate covers, with an exact hopping Hamiltonian on the orbit quotient. We distinguish the symmetry-fixed space from the generally smaller common invariant cyclic envelope generated by the initial state and endpoint Hamiltonians; a prescribed trajectory may span less still. We prove a sector-resolved Schur-complement bound on cover-measurement probability that includes the kinetic floor induced by sector projection. For a retained-walk linear path, a general hopping-cancellation point makes the spectrum exactly orbit-resolved. Multiplicity outside the fixed sector is then dynamically dark, while transverse dark crossings satisfy a separate stability theorem. On even cycles the fixed-sector excitation gap at cancellation equals $\lambda(n-1)/(2n-1)$ despite global multiplicity closure. Separately, for the same instances, we construct a distinct Johnson--Metropolis parent path on the injective hard-core space, from a Dicke state to a Gibbs-amplitude state. Its cover-failure probability is $O(n^{-5})$ and its cyclic-envelope gap is at least $1024\,n^{-13}$ uniformly along the path, yielding a conditional polynomial adiabatic runtime in an abstract Hamiltonian-access model. We claim no quantum speedup; the result is a protocol-dependent separation of global, symmetry-fixed, cyclic-envelope, and tracked spectra.
Abstract:Which part of a Hamiltonian spectrum is relevant when the initial state and interpolation preserve multiple symmetries? We study this question for a multi-register encoding of fixed-cardinality Set Cover. Register permutations and incidence automorphisms define a joint action whose orbits are unlabelled candidate covers, with an exact hopping Hamiltonian on the orbit quotient. We distinguish the symmetry-fixed space from the generally smaller common invariant cyclic envelope generated by the initial state and endpoint Hamiltonians; a prescribed trajectory may span less still. We prove a sector-resolved Schur-complement bound on cover-measurement probability that includes the kinetic floor induced by sector projection. For a retained-walk linear path, a general hopping-cancellation point makes the spectrum exactly orbit-resolved. Multiplicity outside the fixed sector is then dynamically dark, while transverse dark crossings satisfy a separate stability theorem. On even cycles the fixed-sector excitation gap at cancellation equals $\lambda(n-1)/(2n-1)$ despite global multiplicity closure. Separately, for the same instances, we construct a distinct Johnson--Metropolis parent path on the injective hard-core space, from a Dicke state to a Gibbs-amplitude state. Its cover-failure probability is $O(n^{-5})$ and its cyclic-envelope gap is at least $1024\,n^{-13}$ uniformly along the path, yielding a conditional polynomial adiabatic runtime in an abstract Hamiltonian-access model. We claim no quantum speedup; the result is a protocol-dependent separation of global, symmetry-fixed, cyclic-envelope, and tracked spectra.
20 pages, 2 figures, 4 tables. Revised version clarifies cyclic-envelope versus schedule-dependent reachability; adds a full-Hamiltonian sector-localisation audit, explicit comparison of two protocols, expanded literature positioning, exact-cardinality conventions, and archived code/data at this https URL. Main asymptotic theorem unchanged
arXiv:2608.11503 [quant-ph]
(or arXiv:2608.11503v2 [quant-ph] for this version)
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