Lee-Yang tensors and Hamiltonian complexity

TL;DR

This work develops a systematic framework for Lee-Yang tensors, defined by radii $r$ in which their generating polynomials have no zeros, and studies their implications for quantum states, observables, and many-body Hamiltonians. It proves structural closure properties under contraction, establishes a Perron-Frobenius-type uniqueness for LY(r) Hermitian operators with $r>1$, and shows that LY$(r)$ states with $r>1$ admit quasi-polynomial time preparation on both classical and quantum platforms via Barvinok interpolation. The paper then connects LY tensors to Suzuki–Fisher Hamiltonians and EPR-like models, providing numerical evidence that ground states have LY radii at least $r=1/\sqrt{s}$ with spectral gaps at least $1-s^2$, and it discusses implications for efficient quantum adiabatic algorithms for quantum Max-Cut on bipartite graphs. Open questions focus on breaking the $r>1$ barrier, identifying broader Hamiltonians with LY properties, and clarifying the computational complexity landscape of these zero-free regions in quantum many-body systems.

Abstract

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.

Figures (6)

• Figure 1: Plots from Numerical Study \ref{['num:ly_gap']}. Zeros of equatorial polynomials which lie near the disk $\mathbb{D}_{s^{-1/2}}$ for a path and cycle with $10$ vertices and $s=0.5,0.6$ respectively.
• Figure 2: Plots from Numerical Study \ref{['num:sgap']}. For each tree on $n=16,17$ vertices, the spectral gap is plotted as a red point for $100$ uniformly spaced $s\in(0,1)$.
• Figure 3: Plots from Numerical Study \ref{['num:sgap']}. Spectral gap of 1D chains with $3\le n\le 100$ qubits and open boundary conditions. The lower bound $1-s^2$ is achieved at $n=3$ (when the chain is the star graph). Horizontal dashed lines on both plots show a fit $g(s) = 3.03 s (1-s)^{0.609}$ which corresponds to the large $n$ limit (parity of $n$ does not matter in this limit).
• Figure 4: Plots from Numerical Study \ref{['num:sgap_theta']}. For each connected graph on $7,8$ vertices, random phase shifts $\theta_{p,q}$ are sampled and the spectral gap of $H_s(\theta)$ is plotted as a red dot for $100$ uniformly spaced $s\in(0,1)$.
• Figure 5: Complexity landscape for XXZ Heisenberg Hamiltonians. Ground energy problems for models in the pink region are QMA-hard, while those in the grey and white regions are contained in StoqMA piddock2015complexity. Models in the yellow region have the trivial ground state $|0^n\rangle$ while the partition function admits a quasipolynomial time approximation algorithm in the presence of a uniform magnetic field harrow2020classical. The green line represents EPR Hamiltonians. Assuming Conjecture \ref{['conj:sgap']} of Section \ref{['subs:EPRgap']} is true, the ground energy problem on the green line is in BQP in the uniformly weighted case ($w_{ij}=1$ for all $i,j$ in Eq. \ref{['app_XXZ_eq1']}). The models on the blue line admit efficient randomized approximation bravyi2016polynomial. When $f>0$ the partition functions of these models can be approximated by the partition function of the six-vertex model. On the blue line one obtains a six vertex model in a parameter regime where it admits efficient approximation cai2019approximability. In contrast, the grey region maps to a six-vertex model in a parameter regime where it is likely intractable to approximate cai2019approximability. The white region (including the boundaries shown in black and green) maps to a six-vertex model in a parameter regime where its complexity is a mystery.

Theorems & Definitions (48)

• Definition 1
• Lemma 1: Tensor contraction
• Theorem 1: Hurwitz's Theorem
• Corollary 1: Limit of Lee-Yang tensors
• proof
• Lemma 2
• proof
• Theorem 2: Suzuki-Fisher 1971 sf71
• Lemma 3
• proof