利用引导态估计基态能量的最优下界
Optimal Lower Bound for Ground-State Energy Estimation with a Guiding State
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中文总结 AI 辅助
本文针对引导哈密顿问题，证明了基态能量估计的联合下界Ω(log(1/ε)/γδ)，并将其拓展至多种特殊情况，还推导了平方和形式哈密顿量下的对应下界。
中文摘要 AI 辅助
引导哈密顿问题如下：给定对某个哈密顿量H对应的幺正变换U=e^{iH}的访问权限，以及对一个可制备引导态的幺正变换的访问权限，该引导态与H的基态空间的重叠度至少为γ>0，要求估计H的基态能量，使其加性误差δ>0，成功概率至少为1−ε（ε>0）。需要多少次应用U及其逆U⁻¹？此前已知一个上界为O(log(1/ε)log(1/γ)/γδ)，且近期被改进为O(log(1/ε)/γδ) [JW26]。当δ、γ、ε三个参数中某一个保持恒定时，已存在匹配的下界 [MdW26]。本文证明了联合下界Ω(log(1/ε)/γδ)，且当H的维度至少为log(1/ε)/γ²时，该下界具有紧的ε依赖关系。此外，本文表明，对于两种特殊情况，该下界（维度要求略大）同样成立：第一种是基态保证唯一且H的第一与第二本征值之间的间隙为δ；第二种是基态制备，其中δ表示谱间隙，ε现在为近似误差。这些下界也适用于可通过块编码访问哈密顿量的情况，以及允许使用U的分数次幂的情况，如连续时间哈密顿模拟。最后，当H非负且表示为平方和时，已知存在改进的上界，本文的结果对这种情况隐含了下界Ω(log(1/ε)/γ√δ)。
英文摘要
Suppose we can apply the unitary $U=e^{i H}$ for some Hamiltonian $H$, and are given access to a unitary that prepares a guiding state promised to have overlap at least $γ>0$ with the ground space of $H$. Our goal is to estimate the ground-state energy of $H$ within additive error $δ> 0$ and success probability at least $1-\varepsilon$, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? This quantity corresponds to the total Hamiltonian-simulation time needed. An upper bound $O(\log(1/\varepsilon)\log(1/γ)/γδ)$ was known, and was improved to $O(\log(1/\varepsilon)/γδ)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $δ,γ,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $Ω(\log(1/\varepsilon)/γδ)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/γ^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $δ$ between its first and second eigenvalue; and for ground-state preparation, where $δ$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $Ω(\log(1/\varepsilon)/γ\sqrtδ)$ for this case.
发表机构
- Google Quantum AI（谷歌量子人工智能）
- QuSoft, CWI and University of Amsterdam（QuSoft、荷兰数学与计算机科学研究中心和阿姆斯特丹大学）
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