发表机构
Tsinghua University; Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; Center on Frontiers of Computing Studies, Peking University; School of Computer Science, Peking University（清华大学; 中国科学院软件研究所; 中国科学院大学; 北京大学前沿计算研究中心; 北京大学计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了基态制备的最优查询复杂度，提出了两种算法分别达到期望和最坏情况下的最优查询次数，并证明了匹配的下界，同时给出了对态制备预言机的调用界限及其最优性条件。
AI 中文摘要
我们确定了在已知谱隙中能量阈值的情况下，基态制备到迹距离误差 $\varepsilon$ 的最优查询复杂度。设 $U_H$ 是哈密顿量的 $\alpha$-块编码，该哈密顿量具有唯一基态 $|\psi_0\rangle$，并假设对于态制备预言机 $U_I$，有 $|\langle\psi_0|U_I|0\rangle|\ge\gamma$。阈值位于基态能量之上至少 $\Delta/2$ 处，且位于每个激发态能量之下至少 $\Delta/2$ 处。我们给出了两种算法，它们制备的态与基态的迹距离在 $\varepsilon$ 以内。一种算法期望使用 $O((\alpha/\Delta)(\gamma^{-1}+\log(1/\varepsilon)))$ 次对 $U_H$ 的调用；另一种算法在最坏情况下使用 $O((\alpha/(\gamma\Delta))\log(1/\varepsilon))$ 次对 $U_H$ 的调用。我们证明了与期望查询次数匹配的下界；相应的最坏情况下的下界由 Somma 和 de Wolf [SdW26] 得出。对 $U_I$ 的调用次数分别在期望情况下为 $O(1/\gamma)$，在最坏情况下为 $O(\gamma^{-1}\log(1/\varepsilon))$。在 $(N+1)$ 维系统中，当 $U_H$ 调用的期望或最坏情况次数分别为 $o((\alpha/\Delta)\sqrt N)$ 时，这些 $U_I$ 界限也是最优的。两种算法都使用恒定精度的谱滤波器来构造纯化器，然后在振幅放大过程中顺序组合该纯化器，以制备与基态具有恒定重叠的态。期望查询算法重复制备过程并随后进行一次高精度谱滤波，直到成功。最坏情况算法使用精度递增的滤波器，并限制总查询次数。
英文摘要
We determine the optimal query complexity of ground-state preparation to trace-distance error $\varepsilon$ when an energy threshold in the spectral gap is known. Let $U_H$ be an $α$-block-encoding of a Hamiltonian with unique ground state $|ψ_0\rangle$, and suppose $|\langleψ_0|U_I|0\rangle|\geγ$ for a state-preparation oracle $U_I$. The threshold lies at least $Δ/2$ above the ground-state energy and at least $Δ/2$ below every excited-state energy. We give two algorithms that prepare a state within trace distance $\varepsilon$ of the ground state. One uses $O((α/Δ)(γ^{-1}+\log(1/\varepsilon)))$ calls to $U_H$ in expectation; the other uses $O((α/(γΔ))\log(1/\varepsilon))$ calls to $U_H$ in the worst case. We prove a lower bound matching the expected query count; the corresponding worst-case lower bound follows from Somma and de Wolf [SdW26]. The respective bounds on calls to $U_I$ are $O(1/γ)$ in expectation and $O(γ^{-1}\log(1/\varepsilon))$ in the worst case. On $(N+1)$-dimensional systems, these $U_I$ bounds are also optimal when the expected or worst-case count of $U_H$ calls, respectively, is $o((α/Δ)\sqrt N)$. Both algorithms use a constant-accuracy spectral filter to construct a purifier, which we then sequentially compose during amplitude amplification to prepare a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by one high-accuracy spectral filter until success. The worst-case algorithm uses filters of increasing accuracy and limits the total number of queries.
Comments48 pages
