Orbit-resolved spectra and dynamical accessibility in multi-register covering Hamiltonians
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 $λ(n-1)/(2n-1)$ 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.
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BibTeX
@misc{luiz2026orbitresolved,
title = {{Orbit-resolved spectra and dynamical accessibility in multi-register covering Hamiltonians}},
author = {Fabricio de Souza Luiz},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.11503},
eprint = {2608.11503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11503},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orbit-resolved spectra and dynamical accessibility in multi-register covering Hamiltonians
AU - Fabricio de Souza Luiz
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.11503
UR - https://arxiv.org/abs/2608.11503
AB - 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 $λ(n-1)/(2n-1)$ 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.
ER -