发表机构
Global Technology Applied Research, JPMorganChase; CNRS, Université Paris Cité, IRIF（摩根大通全球技术应用研究; 法国国家科学研究中心，巴黎西岱大学，IRIF）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过傅里叶秩方法，为高精度凸优化建立了近最优的量子查询下界，解决了开放问题，并扩展了行列式与最小特征值的下界。
AI 中文摘要
我们在一个显式的$n$维椭球体族上，为高精度凸优化建立了近线性的量子查询下界。我们关注具有显式给定目标的线性优化，其中可行集通过成员资格预言机访问。我们证明，任何算法，对于每个单位线性目标，返回一个精确可行点且加性目标误差为$\Theta(n^{-2})$，需要$\Omega\\!\left(\frac{n}{\log n\\,\log\log n}\right)$次成员资格查询。如果返回点仅需近似可行，在距离可行集$\Theta(n^{-2})$以内，同样的下界也成立。这解决了Chakrabarti、Childs、Li和Wu（《Quantum》，2020年）以及van Apeldoorn、Gilyén、Gribling和de Wolf（《Quantum》，2020年）提出的一个开放问题，直至对数因子。结合这些论文中的上界，高精度凸优化的查询复杂度在直至对数因子的意义上被紧密刻画。该证明围绕行列式计算的下界构建，该下界通过一种基于傅里叶秩的新颖多项式方法推导得出。在连续矩阵相位查询模型中，计算一个实数$n\times n$矩阵的行列式至少需要$n/2$次矩阵-向量乘积查询。该构造还产生了$\Omega(n)$的相位查询下界，用于估计实数对称$n\times n$矩阵的最小特征值，加性精度为$\Theta(n^{-2})$。这些结果将Childs、Hung和Li（ICALP 2021）的行列式和最小特征值下界从有限域扩展到实值情形。基于相同的构造，我们还证明了对于光滑且强凸函数的常数精度优化，存在近最优的梯度查询下界。
英文摘要
We establish a linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $Θ(n^{-1/2})$ requires $Ω(n)$ membership queries, where the ellipsoids are contained in the unit ball. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $Θ(n^{-3/2})$ distance from the feasible set. This resolves an open question posed by Chakrabarti, Childs, Li, and Wu~(\textit{Quantum}, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(\textit{Quantum}, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $Ω(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $Θ(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove an $Ω(\min\{\sqrtκ,n\})$ gradient-query lower bound for constant-accuracy optimization of $1$-strongly convex and $κ$-smooth functions, which is optimal up to a logarithmic factor.
CommentsUpdates from Version 1: Updated proof removing logarithmic factors from lower bound, detailed comparison to independent concurrent work, minor clarifications
