stateQIP = statePSPACE
stateQIP = statePSPACE
摘要：
复杂性理论传统上研究解决经典计算问题的难度。在量子环境中，考虑不同的复杂性概念也是很自然的，即物理上准备某个量子态的复杂性。我们研究了两个这样的状态复杂性类之间的关系：statePSPACE，其中包含的状态，可以产生的空间均匀的多项式空间量子线路，和stateQIP，其中包含的状态，一个多项式时间的量子验证器可以通过与一个全能的不可信的量子证明器交互产生。后一类最近由Rosenthal和Yuen（ITCS 2022）引入，他们证明了statePSPACE $\subseteq$ stateQIP。我们的主要结果是反向包含，stateQIP $\subseteq$ statePSPACE，从而建立了两个类的平等，并提供了一个自然的状态复杂性类似于著名的QIP = PSPACE定理的Jain，等。（J. ACM 2011）。为了证明这一点，我们开发了一个多项式空间的量子算法来解决一个大类的指数大的“PSPACE可计算”半定程序（SDP），这也准备了一个优化编码的量子态。我们的SDP求解器依赖于量子算法的最新块编码技术，这表明这些技术对复杂性理论也很有用。使用类似的技术，我们还表明，一般量子交互协议的最佳证明策略可以在量子多项式空间中实现。我们证明了这一点，通过研究的算法版本的乌尔曼定理，并建立一个上限的复杂性实现乌尔曼变换。
英文摘要：
Complexity theory traditionally studies the hardness of solving classical computational problems. In the quantum setting, it is also natural to consider a different notion of complexity, namely the complexity of physically preparing a certain quantum state. We study the relation between two such state complexity classes: statePSPACE, which contains states that can be generated by space-uniform polynomial-space quantum circuits, and stateQIP, which contains states that a polynomial-time quantum verifier can generate by interacting with an all-powerful untrusted quantum prover. The latter class was recently introduced by Rosenthal and Yuen (ITCS 2022), who proved that statePSPACE $\subseteq$ stateQIP. Our main result is the reverse inclusion, stateQIP $\subseteq$ statePSPACE, thereby establishing equality of the two classes and providing a natural state-complexity analogue to the celebrated QIP = PSPACE theorem of Jain, et al. (J. ACM 2011). To prove this, we develop a polynomial-space quantum algorithm for solving a large class of exponentially large "PSPACE-computable" semidefinite programs (SDPs), which also prepares an optimiser encoded in a quantum state. Our SDP solver relies on recent block-encoding techniques from quantum algorithms, demonstrating that these techniques are also useful for complexity theory. Using similar techniques, we also show that optimal prover strategies for general quantum interactive protocols can be implemented in quantum polynomial space. We prove this by studying an algorithmic version of Uhlmann's theorem and establishing an upper bound on the complexity of implementing Uhlmann transformations.
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。
