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arXiv:2610.09485

The Geometry of Lee-Yang Tensors

Jake Hofgard

quant-phmath-phmath.CVSubmitted Oct 7, 2026• Announced Oct 8, 2026• 0 citations
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Abstract

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 $β\geq 0$.

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43 pages, 1 figure

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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 222-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 kkk-local Hamiltonians, k≥3k \geq 3k≥3, cannot have Lee-Yang radius at least one for all inverse temperatures β≥0β\geq 0β≥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≥0β\geq 0β≥0.
A complex tensor with nnn binary indices can be identified with a multilinear polynomial in nnn complex variables. We say it is a Lee-Yang tensor with radius rrr if the polynomial is nonzero whenever all variables lie in the open disk of radius rrr. 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>1r > 1r>1 can be prepared by quasipolynomial-sized circuits. We also show that every Hermitian operator with Lee-Yang radius r>1r > 1r>1 has a unique principal eigenvector. These results suggest that r=1r = 1r=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⟩+s∣11⟩|00\rangle + s|11\rangle∣00⟩+s∣11⟩ for some 0≤s≤10 \leq s \leq 10≤s≤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/sr = 1/\sqrt{s}r=1/s​ while the spectral gap between the two smallest eigenvalues is at least 1−s21-s^21−s2. 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.
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