Spectral gap of Lee-Yang Hamiltonians

Abstract

The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.

The paper asks whether the Lee–Yang zero-freeness property can imply a quantitative ground-state spectral gap and efficient quantum ground-energy estimation. It considers two-local qubit Hamiltonians on arbitrary graphs with ferromagnetic $ZZ$ couplings satisfying a bondwise dominance condition over $XX$, $YY$, $XY$, and $YX$ interactions. Without cross couplings, this condition reduces to $w^z_{ij}\geq\max(|w^x_{ij}|,|w^y_{ij}|)$. For every Hamiltonian in this class and every uniform field $h>0$, it proves that $H_h=H-h\sum_i Z_i$ has a unique ground state and gap $E_1-E_0\geq h/4$, independently of system size and coupling strengths. The proof retains separate field variables at every imaginary-time Trotter layer. Lee–Yang zero-freeness makes the log-partition function analytic, while a Taylor-support lemma forces contributions connecting distant layers to have high degree. Analytic bounds then yield decay of Trotterized correlations. An adaptive Trotter step controls the continuum limit, establishing connected imaginary-time correlation decay for every $Z_S=\prod_{i\in S}Z_i$ at rate $h/2$, with polynomial prefactors. A spectral representation and the completeness of diagonal Pauli products imply the gap and uniqueness. The gap enables polynomial-time adiabatic ground-energy estimation to inverse-polynomial additive accuracy, placing EPR-Hamiltonian energy estimation and bipartite quantum Max-Cut in $\mathsf{BQP}$. The theorem requires a nonzero field and gives no gap bound at $h=0$; energy estimation there uses a small-field approximation. No classical hardness or quantum advantage is established, and containment of the sign-problem-free EPR case in $\mathsf{BPP}$ remains possible.