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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• Revised Sep 24, 2026 (v2)

v1 · Sep 13, 2026 v2 · Sep 24, 2026

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.

Comments: 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

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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

2μ

, where

μ

is the magnetic field along the

Z

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.

We provide a geometric description of Lee-Yang tensors, building upon several recent results of Bravyi, Gosset, Liu, and Wong. By applying tools from complex Perron-Frobenius theory, we generalize one of their key lemmas to bound the spectral ratio of non-normal operators with Lee-Yang radius greater than one. As a direct consequence, we provide a local-to-global principle for bounding spectral gaps of Suzuki-Fisher Hamiltonians, a family of

2

-local Hamiltonians, using only the Lee-Yang radius of each local operator to bound the gap from below. On the other hand, we develop a simple perturbative technique to show that the Gibbs states of

k

-local Hamiltonians,

k \geq 3

, cannot have Lee-Yang radius at least one for all inverse temperatures

β\geq 0

. We apply this same method to conclude that Suzuki-Fisher Hamiltonians are precisely those Hamiltonians whose Gibbs states are Lee-Yang at all inverse temperatures

β\geq 0

.

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.

A complex tensor with

n

binary indices can be identified with a multilinear polynomial in

n

complex variables. We say it is a Lee-Yang tensor with radius

r

if the polynomial is nonzero whenever all variables lie in the open disk of radius

r

. In this work we study quantum states and observables which are Lee-Yang tensors when expressed in the computational basis. We first review their basic properties, including closure under tensor contraction and certain quantum operations. We show that quantum states with Lee-Yang radius

r > 1

can be prepared by quasipolynomial-sized circuits. We also show that every Hermitian operator with Lee-Yang radius

r > 1

has a unique principal eigenvector. These results suggest that

r = 1

is a key threshold for quantum states and observables. Finally, we consider a family of two-local Hamiltonians where every interaction term energetically favors a deformed EPR state

|00\rangle + s|11\rangle

for some

0 \leq s \leq 1

. We numerically investigate this model and find that on all graphs considered the Lee-Yang radius of the ground state is at least

r = 1/\sqrt{s}

while the spectral gap between the two smallest eigenvalues is at least

1-s^2

. We conjecture that these lower bounds hold more generally; in particular, this would provide an efficient quantum adiabatic algorithm for the quantum Max-Cut problem on uniformly weighted bipartite graphs.

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

\mathsf{SU}(2)

invariant, and its Gibbs states undergo an

\mathsf{SU}(2)

symmetry breaking phase transition at inverse temperature

β=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<2

, the gap as a function of number of qubits

n

is

Θ(1)

, while for fixed

β>2

the gap is

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>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

\mathsf{SU}(2)

and

\mathsf{S}_n

, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.