Estimating ground state energies of many-body Hamiltonians is a central task in many areas of quantum physics. In this work, we give quantum algorithms which, given any $k$-body Hamiltonian $H$, compute an estimate for the ground state energy and prepare a quantum state achieving said energy, respectively. Specifically, for any $\varepsilon>0$, our algorithms return, with high probability, an estimate of the ground state energy of $H$ within additive error $\varepsilon M$, or a quantum state with the corresponding energy. Here, $M$ is the total strength of all interaction terms, which in general is extensive in the system size. Our approach makes no assumptions about the geometry or spatial locality of interaction terms of the input Hamiltonian and thus handles even long-range or all-to-all interactions, such as in quantum chemistry, where lattice-based techniques break down. In this fully general setting, the runtime of our algorithms scales as $2^{cn/2}$ for $c<1$, yielding the first quantum algorithms for low-energy estimation breaking a standard square root Grover speedup for unstructured search. The core of our approach is remarkably simple, and relies on showing that an extensive fraction of the interactions can be neglected with a controlled error. What this ultimately implies is that even arbitrary $k$-local Hamiltonians have structure in their low energy space, in the form of an exponential-dimensional low energy subspace.

翻译：估计多体哈密顿量的基态能量是量子物理多个领域的核心任务。在本工作中，我们提出了量子算法，对于任意$k$-体哈密顿量$H$，分别计算其基态能量的估计值并制备达到该能量的量子态。具体而言，对于任意$\varepsilon>0$，我们的算法以高概率返回$H$基态能量的加性误差$\varepsilon M$内的估计值，或具有相应能量的量子态。此处$M$为所有相互作用项的总强度，通常随系统尺度呈广延性增长。我们的方法不对输入哈密顿量相互作用项的几何结构或空间局域性作任何假设，因此可处理甚至长程或全对全相互作用（如量子化学中晶格方法失效的情形）。在此完全普适的设置下，我们算法的运行时间按$2^{cn/2}$缩放（其中$c<1$），首次实现了在低能态估计问题上突破非结构化搜索中标准平方根Grover加速的量子算法。我们方法的核心极为简洁，其关键在于证明在可控误差范围内可忽略广延比例的相互作用项。这最终意味着即使任意$k$-局域哈密顿量在其低能空间中亦具有结构，即存在指数维度的低能子空间。