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arXiv:2609.26802

Generators of stability-preserving semigroups, spectral gaps, and classical ground-state computation

Dongsheng Wei

math.CVmath-phmath.COquant-phSubmitted Sep 13, 2026• Announced Sep 15, 2026• 0 citations
v1 · Sep 13, 2026v2 · Sep 24, 2026
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Abstract

We classify generators of complex stability-preserving semigroups on finite polynomial boxes and prove polynomial-time recognition from Gaussian-rational binary one-site and two-site inputs. Damping bounded below by $μ>0$ gives strict zero-freeness and a sharp, capacity-independent real-part spectral gap, extending the Hermitian $2μ$ gap of Bravyi, Gosset, Liu, and Wong. At bounded degree and local strength, deterministic classical algorithms approximate ground energies and relative amplitudes against stable product states for Suzuki--Fisher Hamiltonians with fixed positive longitudinal fields; field perturbation gives zero-field energy-value polynomial-time approximation schemes for Einstein-Podolsky-Rosen (EPR) and bipartite Quantum MaxCut on bounded-degree unweighted graphs. An all-temperature converse for full Gibbs tensors characterizes Hermitian qubit Hamiltonians, in a fixed basis, as precisely the edgewise phase-rotated Suzuki--Fisher family up to scalars. Together with strict positive-field Gibbs radii, this answers two questions of Wong, Bravyi, Gosset, and Liu. Weighted token-graph spectral concavity resolves the adjacency and signless-Laplacian conjectures of Apte, Parekh, and Sud. An exact certificate disproves the stronger proposed EPR ground-state radius.

Comments
69 pages. Exact verification scripts included as ancillary files

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We consider a broad family of Heisenberg-type quantum spin systems that were studied by Suzuki and Fisher in 1971. This family includes, for example, the Heisenberg antiferromagnet on any bipartite graph. The ground and Gibbs states of these models are Lee-Yang tensors with radius 1: they are associated with multilinear polynomials that possess an extraordinary zero-freeness property inside the unit polydisk in the complex plane. For each Hamiltonian in this family we show that the spectral gap between the first-excited and ground-state energies is lower bounded by 2μ2μ2μ, where μμμ is the magnetic field along the ZZZ direction. Using this result we obtain an efficient quantum adiabatic algorithm for the ground energy of any model in this family. The proof is based on a new inequality that relates the spectral gap of a positive semidefinite operator to its Lee-Yang radius -- a quantitative strengthening of prior work of the authors that may find applications elsewhere.
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The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.
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Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.
The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and SU(2)\mathsf{SU}(2)SU(2) invariant, and its Gibbs states undergo an SU(2)\mathsf{SU}(2)SU(2) symmetry breaking phase transition at inverse temperature β=2β=2β=2. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed β<2β<2β<2, the gap as a function of number of qubits nnn is Θ(1)Θ(1)Θ(1), while for fixed β>2β>2β>2 the gap is Θ(n−1)Θ(n^{-1})Θ(n−1). The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature (β>2β>2β>2) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups SU(2)\mathsf{SU}(2)SU(2) and Sn\mathsf{S}_nSn​, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.