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 |00rangle + s|11rangle 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-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.