Lee-Yang tensors and Hamiltonian complexity

TL;DR

This paper explores Lee-Yang tensors in quantum states and operators, revealing a critical radius threshold at r=1, and investigates their properties and implications for quantum Hamiltonians and algorithms.

Contribution

It introduces the concept of Lee-Yang tensors in quantum states and operators, analyzing their properties, preparation, eigenvectors, and implications for Hamiltonian ground states and quantum algorithms.

Findings

Quantum states with radius r>1 can be prepared efficiently.

Hermitian operators with radius r>1 have unique principal eigenvectors.

Ground state Lee-Yang radius is at least 1/√s, with a spectral gap of at least 1-s².

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.