Estimating the ground state energy of a local Hamiltonian is a central problem in quantum chemistry. In order to further investigate its complexity and the potential of quantum algorithms for quantum chemistry, Gharibian and Le Gall (STOC 2022) recently introduced the guided local Hamiltonian problem (GLH), which is a variant of the local Hamiltonian problem where an approximation of a ground state (which is called a guiding state) is given as an additional input. Gharibian and Le Gall showed quantum advantage (more precisely, BQP-completeness) for GLH with $6$-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to $1/2$ with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.

翻译：估计局域哈密顿量的基态能量是量子化学中的一个核心问题。为进一步探究其复杂性及量子算法在量子化学中的潜力，Gharibian 与 Le Gall（STOC 2022）近期引入了引导式局域哈密顿量问题（GLH）。该问题是局域哈密顿量问题的一个变体，其中基态的近似（称为引导态）作为附加输入给出。Gharibian 与 Le Gall 证明了当引导态与基态的保真度（反多项式地）接近 1/2 时，对于 6-局域哈密顿量，GLH 具有量子优势（更精确地说，是 BQP-完备性）。在本文中，我们优化了局域性和保真度参数：我们证明即使对于 2-局域哈密顿量，且引导态与基态的保真度（反多项式地）接近 1 时，BQP-完备性依然成立。此外，我们还证明对于二维方格或二维三角晶格上受物理启发的 2-局域哈密顿量，BQP-完备性同样成立。除基态能量估计的难度外，我们还证明当考虑估计这些哈密顿量的激发态能量时，BQP-难度依然存在。这些结果进一步推动了在量子化学中建立实用量子优势的进程。