The local Hamiltonian (LH) problem is the canonical $\mathsf{QMA}$-complete problem introduced by Kitaev. In this paper, we show its hardness in a very strong sense: we show that the 3-local Hamiltonian problem on $n$ qubits cannot be solved classically in time $O(2^{(1-\varepsilon)n})$ for any $\varepsilon>0$ under the Strong Exponential-Time Hypothesis (SETH), and cannot be solved quantumly in time $O(2^{(1-\varepsilon)n/2})$ for any $\varepsilon>0$ under the Quantum Strong Exponential-Time Hypothesis (QSETH). These lower bounds give evidence that the currently known classical and quantum algorithms for LH cannot be significantly improved. Furthermore, we are able to demonstrate fine-grained complexity lower bounds for approximating the quantum partition function (QPF) with an arbitrary constant relative error. Approximating QPF with relative error is known to be equivalent to approximately counting the dimension of the solution subspace of $\mathsf{QMA}$ problems. We show the SETH and QSETH hardness to estimate QPF with constant relative error. We then provide a quantum algorithm that runs in $O(\sqrt{2^n})$ time for an arbitrary $1/\mathrm{poly}(n)$ relative error, matching our lower bounds and improving the state-of-the-art algorithm by Bravyi, Chowdhury, Gosset, and Wocjan (Nature Physics 2022) in the low-temperature regime. To prove our fine-grained lower bounds, we introduce the first size-preserving circuit-to-Hamiltonian construction that encodes the computation of a $T$-time quantum circuit acting on $N$ qubits into a $(d+1)$-local Hamiltonian acting on $N+O(T^{1/d})$ qubits. This improves the standard construction based on the unary clock, which uses $N+O(T)$ qubits.

翻译：局部哈密顿量（LH）问题是Kitaev引入的典型$\mathsf{QMA}$完全问题。本文中，我们以极强的形式证明了其困难性：在强指数时间假设（SETH）下，对于任意$\varepsilon>0$，经典计算机无法在$O(2^{(1-\varepsilon)n})$时间内求解$n$个量子比特上的3-局部哈密顿量问题；在量子强指数时间假设（QSETH）下，对于任意$\varepsilon>0$，量子计算机无法在$O(2^{(1-\varepsilon)n/2})$时间内求解该问题。这些下界表明，当前已知的LH经典与量子算法难以实现显著改进。此外，我们能够证明以任意常数相对误差近似量子配分函数（QPF）的细粒度复杂性下界。已知以相对误差近似QPF等价于近似计数$\mathsf{QMA}$问题解空间的维度。我们证明了在SETH与QSETH下，以常数相对误差估计QPF是困难的。随后，我们提出了一种量子算法，对于任意$1/\mathrm{poly}(n)$相对误差，其运行时间为$O(\sqrt{2^n})$，与我们的下界匹配，并在低温区域改进了Bravyi、Chowdhury、Gosset和Wocjan（Nature Physics 2022）的最先进算法。为证明我们的细粒度下界，我们首次提出了规模保持的电路-哈密顿量构造，该构造将作用于$N$个量子比特的$T$时间量子电路计算编码为作用于$N+O(T^{1/d})$个量子比特的$(d+1)$-局部哈密顿量。这改进了基于一元时钟的标准构造（该构造使用$N+O(T)$个量子比特）。