Low-energy estimation and state preparation for general $k$-local Hamiltonians are fundamental challenges in quantum complexity theory. For constant relative accuracy, Buhrman et al. (PRL 2025) recently broke the natural Grover bound $O(2^{n/2})$, where $n$ denotes the number of qubits, for both problems. In this paper, for any sufficiently small parameter $d\ge 0$, we present an even faster quantum algorithm that outputs a quantum state with energy bounded by the minimum energy over all depth-$d$ states (i.e., states obtained by applying a depth-$d$ circuit to the all-zero state), together with an estimate of this energy. For the class of Hamiltonians with depth-$d$ ground states, our algorithm furthermore achieves exactly the same energy guarantees as Buhrman et al. Our results also provide insight into the distinction between strongly entangled states and those admitting efficient classical descriptions.

翻译：对于一般的$k$-局部哈密顿量而言，低能估计和量子态制备是量子复杂性理论中的核心挑战。布赫尔曼等人（PRL 2025）近期在恒定相对精度下，针对这两个问题突破了自然的格罗弗界限$O(2^{n/2})$（其中$n$表示量子比特数）。本文针对任意足够小的参数$d\ge 0$，提出了一种更快的量子算法：该算法能输出一个能量不超过所有深度-$d$态（即对全零态施加深度为$d$的量子电路所得的量子态）中最小能量的量子态，并同时给出该能量的估计值。对于具有深度-$d$基态的哈密顿量族，本算法还能达到与布赫尔曼等人完全相同的能量保证。我们的结果也揭示了强纠缠态与允许有效经典描述的状态之间的区别。