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 $\varOmega(1/\epsilon)$ lower bound for the number of reflections needed to estimate the partition function with a quantum algorithm. The proof is based on a reduction from the problem of estimating the Hamming weight of an unknown binary string.

翻译：我们研究了吉布斯分布配分函数 $\mathsf{Z}(\beta)=\sum_{x\in\chi} e^{-\beta H(x)}$（由哈密顿量 $H(x)$ 表征）的估计复杂性。通过依赖吉布斯态相干编码的反射，我们为求解该任务的量子算法提供了一个简单且自然的下界。主要贡献在于：证明量子算法估计配分函数所需的反射次数下界为 $\varOmega(1/\epsilon)$。该证明基于对未知二进制字符串汉明权重的估计问题的归约。