量子随机性的导数：将伪随机酉算子与伪随机（函数式）态分离

中文摘要 AI 辅助

量子计算催生了新的用于态和酉算子的伪随机原语，包括伪随机态生成器（PRSG）、伪随机函数式态生成器（PRFSG）以及伪随机酉算子（PRU）。在本文中，我们展示了PRFSG与PRU之间的完全酉预言机分离。该分离在最强的态概念与最弱的酉算子概念之间成立：即使是自适应安全的、量子可访问的PRFSG，也不蕴含非自适应安全的、仅前向的PRU，即使允许它们的实现是非酉的并使用任意数量的辅助量子比特。这揭示了量子态与量子酉算子的伪随机性之间的根本区别。我们的主要技术思想是将一个可访问态生成预言机的候选PRU构造视为从底层预言机态到所实现酉算子的映射，并研究该映射的导数。这些导数本质上是低秩的，我们利用这种低秩结构将所得酉算子与真正随机的酉算子区分开来。我们相信这种微分视角可能有助于研究关于量子态和酉算子的其他结构性问题。

英文摘要

Quantum computation gives rise to new pseudorandom primitives for states and unitaries, including pseudorandom state generators (PRSGs), pseudorandom function-like state generators (PRFSGs), and pseudorandom unitaries (PRUs). In this paper, we show a full unitary oracle separation between PRFSGs and PRUs. The separation holds between the strongest state notion and the weakest unitary notion: even adaptively secure, quantum-accessible PRFSGs do not imply non-adaptively secure, forward-only PRUs, even when their implementations are allowed to be non-unitary and use an arbitrary number of ancillary qubits. This reveals a fundamental distinction between pseudorandomness for quantum states and for quantum unitaries. Our main technical idea is to view a candidate PRU construction with access to state generation oracles as a map from the underlying oracle states to implemented unitaries, and to study the derivatives of this map. These derivatives are inherently low rank, and we exploit this low-rank structure to distinguish the resulting unitaries from truly random ones. We believe this differential perspective may be useful for studying other structural questions about quantum states and unitaries.

发表机构

• KAIST（韩国科学技术院）

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