Lee-Yang tensors and Hamiltonian complexity

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.

💡 Research Summary

The paper introduces “Lee‑Yang tensors” as a class of complex tensors whose associated multivariate polynomial is non‑zero whenever all variables lie inside an open disk of a given radius r. By identifying an n‑qubit state, density matrix, operator or channel with such a tensor, the authors bring the classical Lee‑Yang theorem from statistical mechanics into quantum information theory.

First, they formalize the notion: for a tensor ψ with entries ψ_x (x∈{0,1}ⁿ) the generating polynomial is f_ψ(z)=∑x ψ_x ∏{a∈supp(x)} z_a. ψ is a Lee‑Yang tensor of radius r if f_ψ(z)≠0 for all z∈D_r (the open polydisk of radius r). They prove that this class is closed under tensor product and under contraction of any pair of indices whose radii multiply to at least 1 (Lemma 1). Consequently, any tensor network built from Lee‑Yang tensors with r≥1 yields a final tensor that is also Lee‑Yang. Using Hurwitz’s theorem they show that limits of sequences of Lee‑Yang tensors remain Lee‑Yang.

Next, they study the effect of measurements and channels. Lemma 3 shows that post‑selected equatorial measurements (including X‑ or Y‑basis measurements) map a Lee‑Yang state with r≥1 either to another Lee‑Yang state of the same radius or to the zero vector. They define a quantum channel to be “in LY(r)” when its Choi matrix is a Lee‑Yang tensor. Theorem 3 characterizes a family of single‑qubit Pauli channels (those with min{p₀,p₃} ≥ max{p₁,p₂}) whose Choi matrices lie in LY(1), implying that such channels preserve the Lee‑Yang property of any input state.

The authors then turn to Hermitian operators. Lemma 4 (a complex‑valued version of the Griffiths inequality) states that any Hermitian operator P∈LY(1) with ⟨0ⁿ|P|0ⁿ⟩>0 has a unique principal eigenvector, and that eigenvector itself belongs to LY(1). This mirrors the Perron‑Frobenius theorem for non‑negative matrices and guarantees a distinguished “non‑negative” direction in the Hilbert space.

A major algorithmic contribution appears in Section 4: any n‑qubit state that is a Lee‑Yang tensor with radius r>1 can be prepared by a quantum circuit of quasipolynomial size. The construction repeatedly applies single‑qubit channels whose Choi matrices are in LY(1) together with appropriate scaling, exploiting the fact that the polynomial never vanishes inside D_r to keep amplitudes bounded away from zero. The resulting circuit depth scales as polylog (n)·poly(1/(r−1)).

Finally, the paper investigates a family of two‑local Hamiltonians whose interaction terms energetically favor the deformed EPR state |ψ_s⟩=|00⟩+s|11⟩ with 0≤s≤1. Each term is proportional to the projector onto |ψ_s⟩, and the full Hamiltonian is a sum over edges of a graph (not necessarily planar). Numerical experiments on a variety of graphs (complete bipartite, lattices, random bipartite) reveal two striking lower bounds: (i) the Lee‑Yang radius of the ground state is at least 1/√s, and (ii) the spectral gap between the lowest two eigenvalues is at least 1−s². The authors conjecture that these bounds hold for all bipartite graphs and all s. If true, the ground‑state Gibbs distribution would be zero‑free on a disk of radius ≥1, enabling polynomial‑time approximation of expectation values via Barvinok‑type polynomial interpolation. This would yield an efficient quantum adiabatic algorithm for the quantum Max‑Cut problem on uniformly weighted bipartite graphs—a problem known to be QMA‑hard in general.

In summary, the work bridges a deep result from statistical physics (Lee‑Yang zero‑freeness) with modern quantum complexity theory. It shows that zero‑free multivariate polynomials impose strong algebraic constraints that translate into closure properties for tensor networks, uniqueness of principal eigenvectors, efficient state preparation, and potential algorithmic breakthroughs for hard optimization problems. The paper thus opens a promising new avenue for exploiting analytic properties of quantum tensors in both theoretical and algorithmic contexts.