带量子态的近似计数问题的紧量子下界
Tight Quantum Lower Bound for Approximate Counting with Quantum States
AI总结：
本文完全解决了带量子态资源的近似计数问题，在 $1/k \le \varepsilon \le 1$ 范围内给出所有资源类型之间的紧下界与权衡，并证明下界是紧的，从而闭合了 Aaronson 等人提出的主要未解决问题。
摘要：
我们证明了 Aaronson、Kothari、Kretschmer 和 Thaler（2020）所考虑的计数问题变体的紧下界。任务是区分输入集合 $x\subseteq [n]$ 的大小是 $k$ 还是 $k'=(1+\varepsilon)k$。我们假设算法可以访问：成员关系预言机，对于每个 $i\in [n]$，它能回答 $i$ 是否属于 $x$；以及 $x$ 中元素上的均匀叠加态 $|\psi_x\rangle = \sum_{i\in x} |i\rangle/\sqrt{|x|}$。此外，我们考虑算法访问该状态的三种不同方式：算法可以拥有状态 $|\psi_x\rangle$ 的副本；算法可以执行关于状态 $|\psi_x\rangle$ 的反射预言机；算法可以执行状态生成预言机（或其逆），该预言机执行变换 $|0\rangle\mapsto|\psi_x\rangle$。如果没有第二类资源（与 $|\psi_x\rangle$ 相关的资源），该问题已被充分理解。Aaronson 等人最近发起了对带有第二类资源的问题的研究。我们完全解决了所有 $1/k \le \varepsilon\le 1$ 取值下的问题，给出了算法可用的所有类型资源之间的紧权衡。我们还证明了我们的下界是紧的。因此，我们解决了 Aaronson 等人提出的主要未解决问题。这些下界使用 Belovs（2015）的对手界变体证明，并运用对称群的表示论，将其应用于 $S_n$-模 $\mathbb{C}^{\binom{[n]}k}$ 和 $\mathbb{C}^{\binom{[n]}k}\otimes \mathbb{C}$。
英文摘要：
We prove tight lower bounds for the following variant of the counting problem considered by Aaronson, Kothari, Kretschmer, and Thaler (2020). The task is to distinguish whether an input set $x\subseteq [n]$ has size either $k$ or $k'=(1+\varepsilon)k$. We assume the algorithm has access to * the membership oracle, which, for each $i\in [n]$, can answer whether $i\in x$, or not; and \item the uniform superposition $|ψ_x\rangle = \sum_{i\in x} |i\rangle/\sqrt{|x|}$ over the elements of $x$. Moreover, we consider three different ways how the algorithm can access this state: - the algorithm can have copies of the state $|ψ_x\rangle$; - the algorithm can execute the reflecting oracle which reflects about the state $|ψ_x\rangle$; - the algorithm can execute the state-generating oracle (or its inverse) which performs the transformation $|0\rangle\mapsto|ψ_x\rangle$. Without the second type of resources (the ones related to $|ψ_x\rangle$), the problem is well-understood. The study of the problem with the second type of resources was recently initiated by Aaronson et al. We completely resolve the problem for all values of $1/k \le \varepsilon\le 1$, giving tight trade-offs between all types of resources available to the algorithm. We also demonstrate that our lower bounds are tight. Thus, we close the main open problems from Aaronson et al. The lower bounds are proven using variants of the adversary bound from Belovs (2015) and employing representation theory of the symmetric group applied to the $S_n$-modules $\mathbb{C}^{\binom{[n]}k}$ and $\mathbb{C}^{\binom{[n]}k}\otimes \mathbb{C}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
