实现单消息和双消息量子证明系统的完美完备性
Achieving perfect completeness for one- and two-message quantum proof systems
浏览论文内容
中文总结 AI 辅助
本研究证明单消息和双消息量子证明系统（QMA、QAM、QIP(2) 等）可实现完美完备性，通过块编码矩阵和回合减半变换解决长期开放问题。
中文摘要 AI 辅助
虽然使用至少三条消息的量子交互式证明系统可以实现完美完备性（如 Kitaev 和 Watrous 在 STOC 2000 中所证明），但对于单消息和双消息量子证明系统是否可以实现完美完备性，这一问题一直悬而未决。对于单消息情形，QMA 是否能实现完美完备性在 Watrous (FOCS 2000) 和 Aharonov 与 Naveh (2002) 中被作为开放问题提出；对于双消息情形，相应的问题在 Jain、Upadhyay 和 Watrous (FOCS 2009) 以及 Kobayashi、Le Gall 和 Nishimura (SICOMP, 2019) 中被（隐式地）提出。在本工作中，我们证明了 QIP(2)、qq-QAM、QAM 和 QMA 可以实现完美完备性。这里 qq-QAM 表示允许双消息量子公币量子交互式证明系统的承诺问题类，其中验证者的唯一消息由半 EPR 对组成。我们的主要技术贡献如下：1. 对于 QMA（并直接适用于 QAM），我们构造了一个精确可构造的块编码矩阵，其核可以证明是实例，该矩阵由验证电路诱导的接受算子构造而来。2. 对于 QIP(2)（并隐含地适用于 qq-QAM），我们提出了一种新的回合减半变换，该变换保持完备性，并确保所得证明系统至少保留两条消息，前提是最终测量前的终止状态可以被高效制备。
英文摘要
While quantum interactive proof systems using at least three messages can achieve perfect completeness, as shown by Kitaev and Watrous (STOC 2000), whether perfect completeness is achievable for one- and two-message quantum proof systems has remained open. For the one-message case, whether $\sf QMA$ can achieve perfect completeness was posed as an open problem in Watrous (FOCS 2000) and Aharonov and Naveh (2002); for the two-message case, the corresponding problems were (implicitly) posed in Jain, Upadhyay, and Watrous~(FOCS 2009) and Kobayashi, Le Gall, and Nishimura (SICOMP, 2019). In this work, we establish that ${\sf QIP}(2)$, ${\rm qq}\text{-}{\sf QAM}$, $\sf QAM$, and $\sf QMA$ can achieve perfect completeness. Here ${\rm qq}\text{-}{\sf QAM}$ denotes the class of promise problems admitting two-message quantum-public-coin quantum interactive proof systems in which the verifier's only message consists of half-EPR pairs. Our main technical contributions are the follows: 1. For $\sf QMA$ (and directly for $\sf QAM$), an exactly constructible block-encoded matrix whose kernel certifies yes instances, constructed from the acceptance operator induced by the verification circuit. 2. For ${\sf QIP}(2)$ (and implicitly ${\rm qq}\text{-}{\sf QAM}$), a new turn-halving transformation that preserves completeness and ensures that the resulting proof system retains at least two messages, provided that the terminal state before the final measurement is efficiently preparable.
发表机构
- École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。
