发表机构
National University of Singapore; Royal Holloway University of London; University of Oxford; Nanyang Technological University（新加坡国立大学; 伦敦皇家霍洛威学院; 牛津大学; 南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对分数阶量子演化问题，证明量子奇异值变换的查询复杂度上界 $O(\frac{1}{\delta}\log\frac{1}{\varepsilon})$ 是最优的，并给出两种下界证明方法。
AI 中文摘要
给定对未知酉算子 $U=e^{iH}$ 的预言机访问，分数阶查询问题询问：当谱与分支切割由间隙 $\delta$ 分隔时，实现非整数幂 $U^t=e^{itH}$（$0<t<1$）需要多少次查询。量子奇异值变换给出了近似误差 $\varepsilon$ 下 $O\\!\left(\frac{1}{\delta}\log\frac{1}{\varepsilon}\right)$ 次查询的上界。我们证明了任意查询算法的匹配下界。我们的论证将任何 $N$ 次查询电路归结为用次数受 $O(N)$ 限制的三角多项式逼近 $e^{it\theta}$，并结合 Remez 不等式。这使我们能够建立 $\Omega_\tau\\!\left(\frac{1}{\delta}\log\frac{1}{\varepsilon}\right)$ 的下界。因此，分数阶查询问题的最优查询复杂度为 $\Theta_{\tau}\\!\left(\frac{1}{\delta}\log\frac{1}{\varepsilon}\right)$，表明已知的 QSVT 构造是渐近最优的。我们还给出了另一种下界证明，基于构造一个湮灭逼近空间的线性泛函，产生一个对 $\delta$ 一致的 $\Omega_{\tau}\\!\left(\log\frac{1}{\varepsilon}\right)$ 界。
英文摘要
Given oracle access to an unknown unitary $U=e^{iH}$ , the fractional query problem asks how many queries are required to implement a noninteger power $U^t=e^{itH}$, $0<t<1$, when the spectrum is separated from the branch cut by a gap $δ$. Quantum singular value transformation gives an upper bound of $O\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$ queries for approximation error $\varepsilon$. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any $N$-query circuit to the approximation of $e^{itθ}$ by a trigonometric polynomial with degree bounded by $O(N)$, together with Remez inequality. This allows us to establish the lower bound of $Ω_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$. Consequently, the optimal query complexity for fractional query problem is $Θ_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$, showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a $Ω_τ\!\left(\log\frac{1}{\varepsilon}\right)$ bound uniform to $δ$.
