We study the complexity of estimating the partition function ${\mathsf{Z}}(\beta)=\sum_{x\in\chi} e^{-\beta H(x)}$ for a Gibbs distribution characterized by the Hamiltonian $H(x)$. We provide a simple and natural lower bound for quantum algorithms that solve this task by relying on reflections through the coherent encoding of Gibbs states. Our primary contribution is a $\Omega(1/\epsilon)$ lower bound for the number of reflections needed to estimate the partition function with a quantum algorithm. We also prove a $\Omega(1/\epsilon^2)$ query lower bound for classical algorithms. The proofs are based on a reduction from the problem of estimating the Hamming weight of an unknown binary string.

翻译：我们研究由哈密顿量 $H(x)$ 刻画的吉布斯分布配分函数 ${\mathsf{Z}}(\beta)=\sum_{x\in\chi} e^{-\beta H(x)}$ 的估计复杂性。我们基于吉布斯态的相干编码反射，为求解此任务的量子算法提供了一个简单而自然的下界。主要贡献是证明了量子算法估计配分函数所需的反射次数下界为 $\Omega(1/\epsilon)$。我们还证明了经典算法的 $\Omega(1/\epsilon^2)$ 查询下界。证明基于对未知二进制字符串汉明权重估计问题的归约。