海森堡自旋系统的高效量子算法
Efficient quantum algorithm for Heisenberg spin systems
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中文总结 AI 辅助
研究海森堡型量子自旋系统，通过证明其谱隙下限，基于新不等式获得该家族模型基态能量的高效量子绝热算法，强化了先前工作成果。
中文摘要 AI 辅助
我们考虑了铃木和费舍尔在1971年研究的一类广泛的海森堡型量子自旋系统。该家族包括例如任何二分图上的海森堡反铁磁体。这些模型的基态和吉布斯态是半径为1的李 - 杨张量：它们与复平面上单位多圆盘内具有非凡无零性的多元多项式相关。对于该家族中的每个哈密顿量，我们表明第一激发态和基态能量之间的谱隙下限为$2\mu$，其中$\mu$是沿$Z$方向的磁场。利用这一结果，我们获得了该家族中任何模型基态能量的高效量子绝热算法。证明基于一个新的不等式，该不等式将正半定算子的谱隙与其李 - 杨半径相关联，这是作者先前工作的定量强化，可能在其他地方有应用。
英文摘要
We consider a broad family of Heisenberg-type quantum spin systems that were studied by Suzuki and Fisher in 1971. This family includes, for example, the Heisenberg antiferromagnet on any bipartite graph. The ground and Gibbs states of these models are Lee-Yang tensors with radius 1: they are associated with multilinear polynomials that possess an extraordinary zero-freeness property inside the unit polydisk in the complex plane. For each Hamiltonian in this family we show that the spectral gap between the first-excited and ground-state energies is lower bounded by $2μ$, where $μ$ is the magnetic field along the $Z$ direction. Using this result we obtain an efficient quantum adiabatic algorithm for the ground energy of any model in this family. The proof is based on a new inequality that relates the spectral gap of a positive semidefinite operator to its Lee-Yang radius -- a quantitative strengthening of prior work of the authors that may find applications elsewhere.
