aiqbench analysis
This paper establishes a rigorous mathematical framework linking stability-preserving semigroups to deterministic classical algorithms for computing ground energies of specific quantum Hamiltonians. By extending spectral gap results to complex generators, it provides new approximation schemes for problems like Quantum MaxCut and characterizes the precise conditions under which Gibbs tensors remain zero-free. The work directly addresses open questions regarding the relationship between Lee-Yang zeros and Suzuki-Fisher families in quantum physics.
Generators of stability-preserving semigroups, spectral gaps, and classical ground-state computation
arXiv:2609.26802v2 Announce Type: cross Abstract: We classify the generators of stability-preserving semigroups on polynomial spaces with bounded coordinate degrees. The generators have differential order at most two, with principal coefficients characterized by low-degree nonnegativity conditions. In disk coordinates, positive degree damping gives strict zero-freeness and a sharp spectral gap, extending the Hermitian gap of Bravyi, Gosset, Liu, and Wong to complex generators. The resulting analytic domain yields deterministic classical algorithms for ground energies and stable product-state queries for bounded-degree Suzuki-Fisher Hamiltonians with bounded local strength and fixed positive fields. Field perturbation gives zero-field energy-value approximation schemes for bounded-degree unweighted EPR and bipartite Quantum MaxCut. We also prove that, up to scalars, the Hermitian qubit Hamiltonians whose full Gibbs tensors are Lee-Yang at every temperature in a fixed basis are precisely the edgewise phase-rotated Suzuki-Fisher family. With strictly positive longitudinal fields, these tensors are zero-free on a polydisk of radius greater than one at each positive inverse temperature. These two statements answer questions of Wong, Bravyi, Gosset, and Liu. Degree-preserving, coefficient-positive stability semigroups have concave sector growth rates, yielding token-graph concavity. We also give a counterexample to a proposed explicit ground-state radius.
Abstract:We classify the generators of stability-preserving semigroups on polynomial spaces with bounded coordinate degrees. The generators have differential order at most two, with principal coefficients characterized by low-degree nonnegativity conditions. In disk coordinates, positive degree damping gives strict zero-freeness and a sharp spectral gap, extending the Hermitian gap of Bravyi, Gosset, Liu, and Wong to complex generators. The resulting analytic domain yields deterministic classical algorithms for ground energies and stable product-state queries for bounded-degree Suzuki-Fisher Hamiltonians with bounded local strength and fixed positive fields. Field perturbation gives zero-field energy-value approximation schemes for bounded-degree unweighted EPR and bipartite Quantum MaxCut. We also prove that, up to scalars, the Hermitian qubit Hamiltonians whose full Gibbs tensors are Lee-Yang at every temperature in a fixed basis are precisely the edgewise phase-rotated Suzuki-Fisher family. With strictly positive longitudinal fields, these tensors are zero-free on a polydisk of radius greater than one at each positive inverse temperature. These two statements answer questions of Wong, Bravyi, Gosset, and Liu. Degree-preserving, coefficient-positive stability semigroups have concave sector growth rates, yielding token-graph concavity. We also give a counterexample to a proposed explicit ground-state radius.
64 pages. Substantially reorganized and rewritten for clarity; references updated, including independent work by Jiang (arXiv:2609.25037). Mathematical results unchanged; theorem numbering changed. Exact verification scripts included as ancillary files
Complex Variables (math.CV); Mathematical Physics (math-ph); Combinatorics (math.CO); Quantum Physics (quant-ph)
arXiv:2609.26802 [math.CV]
(or arXiv:2609.26802v2 [math.CV] for this version)
Full text under the publisher's open licence · 234 words
