We investigate the computational complexity of the Local Hamiltonian (LH) problem and the approximation of the Quantum Partition Function (QPF), two central problems in quantum many-body physics and quantum complexity theory. Both problems are known to be QMA-hard, and under the widely believed assumption that $\mathsf{BQP} \neq \mathsf{QMA}$, no efficient quantum algorithm exits. The best known quantum algorithm for LH runs in $O\bigl(2^{\frac{n}{2}(1 - o(1))}\bigr)$ time, while for QPF, the state-of-the-art algorithm achieves relative error $\delta$ in $O^\ast\bigl(\frac{1}{\delta}\sqrt{\frac{2^n}{Z}}\bigr)$ time, where $Z$ denotes the value of the partition function. A nature open question is whether more efficient algorithms exist for both problems. In this work, we establish tight conditional lower bounds showing that these algorithms are nearly optimal. Under the plausible Quantum Strong Exponential Time Hypothesis (QSETH), we prove that no quantum algorithm can solve either LH or approximate QPF significantly faster than $O(2^{n/2})$, even for 3-local Hamiltonians. In particular, we show: 1) 3-local LH cannot be solved in time $O(2^{\frac{n}{2}(1-\varepsilon)})$ for any $\varepsilon > 0$ under QSETH; 2) 3-local QPF cannot be approximated up to any constant relative error in $O(2^{\frac{n}{2}(1-\varepsilon)})$ time for any $\varepsilon > 0$ under QSETH; and 3) we present a quantum algorithm that approximates QPF up to relative error $1/2 + 1/\mathrm{poly}(n)$ in $O^\ast(2^{n/2})$ time, matching our conditional lower bound. Notably, our results provide the first fine-grained lower bounds for both LH and QPF with fixed locality. This stands in sharp contrast to QSETH and the trivial fine-grained lower bounds for LH, where the locality of the SAT instance and the Hamiltonian depends on the parameter $\varepsilon$ in the $O(2^{\frac{n}{2}(1-\varepsilon)})$ running time.

翻译：我们研究了局域哈密顿量（LH）问题的计算复杂度与量子配分函数（QPF）的近似问题，这两个问题是量子多体物理与量子复杂度理论中的核心问题。已知这两个问题都是QMA-困难的，且在广泛接受的假设$\mathsf{BQP} \neq \mathsf{QMA}$下，不存在高效的量子算法。目前已知的LH最佳量子算法运行时间为$O\bigl(2^{\frac{n}{2}(1 - o(1))}\bigr)$，而对于QPF，最先进的算法可在$O^\ast\bigl(\frac{1}{\delta}\sqrt{\frac{2^n}{Z}}\bigr)$时间内达到相对误差$\delta$，其中$Z$表示配分函数的值。一个自然的开放问题是：对于这两个问题是否存在更高效的算法？在本工作中，我们建立了严格的条件性下界，表明这些算法近乎最优。在合理的量子强指数时间假设（QSETH）下，我们证明即使对于3-局域哈密顿量，任何量子算法也无法以显著快于$O(2^{n/2})$的速度求解LH或近似QPF。具体而言，我们证明：1）在QSETH下，对于任意$\varepsilon > 0$，3-局域LH无法在$O(2^{\frac{n}{2}(1-\varepsilon)})$时间内求解；2）在QSETH下，对于任意$\varepsilon > 0$，3-局域QPF无法在$O(2^{\frac{n}{2}(1-\varepsilon)})$时间内以任意恒定相对误差近似；以及3）我们提出了一种量子算法，可在$O^\ast(2^{n/2})$时间内以$1/2 + 1/\mathrm{poly}(n)$的相对误差近似QPF，与我们的条件性下界相匹配。值得注意的是，我们的结果为具有固定局域性的LH和QPF提供了首个细粒度下界。这与QSETH以及LH的平凡细粒度下界形成鲜明对比，在后两者中，SAT实例和哈密顿量的局域性取决于$O(2^{\frac{n}{2}(1-\varepsilon)})$运行时间中的参数$\varepsilon$。