量子多项式层级：Karp-Lipton定理、错误约化与下界
Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds
AI总结：
本文研究量子多项式层级的三种基于验证者的推广，解决QCPH的坍缩与Karp-Lipton定理，提出pureQPH类并证明其单侧错误约化及包含关系。
摘要：
多项式时间层级（$\mathsf{PH}$）是经典复杂性理论的核心内容，其应用范围涵盖随机化计算、电路下界以及近期量子计算机的“量子优势”分析。然而，在量子领域，尽管至少存在四种量子$\mathsf{PH}$的定义，但即使要为这些定义证明来自$\mathsf{PH}$的基本事实的类似结论也一直具有挑战性。本工作研究了三种基于量子验证者的$\mathsf{PH}$推广，其中两种来自[Gharibian, Santha, Sikora, Sundaram, Yirka, 2022]，分别使用经典字符串（$\mathsf{QCPH}$）和量子混合态（$\mathsf{QPH}$）作为证明，另一种是本文新提出的，使用量子纯态（$\mathsf{pureQPH}$）作为证明。我们首先解决了[GSSSY22]中的若干开放问题，包括$\mathsf{QCPH}$的坍缩定理和Karp-Lipton定理。然后，对于我们新提出的类$\mathsf{pureQPH}$，我们证明了$\mathsf{pureQPH}$的单侧错误约化，并首次给出了这些$\mathsf{PH}$的量子变体之间的关系界限，即$\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}$。
英文摘要：
The Polynomial-Time Hierarchy ($\mathsf{PH}$) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum $\mathsf{PH}$ exist, it has been challenging to prove analogues for these of even basic facts from $\mathsf{PH}$. This work studies three quantum-verifier based generalizations of $\mathsf{PH}$, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings ($\mathsf{QCPH}$) and quantum mixed states ($\mathsf{QPH}$) as proofs, and one of which is new to this work, utilizing quantum pure states ($\mathsf{pureQPH}$) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for $\mathsf{QCPH}$. Then, for our new class $\mathsf{pureQPH}$, we show one-sided error reduction for $\mathsf{pureQPH}$, as well as the first bounds relating these quantum variants of $\mathsf{PH}$, namely $\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}$.
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。
