Quantum pseudorandom state generators (PRSGs) have stimulated exciting developments in recent years. A PRSG, on a fixed initial (e.g., all-zero) state, produces an output state that is computationally indistinguishable from a Haar random state. However, pseudorandomness of the output state is not guaranteed on other initial states. In fact, known PRSG constructions provably fail on some initial state. In this work, we propose and construct quantum Pseudorandom State Scramblers (PRSSs), which can produce a pseudorandom state on an arbitrary initial state. In the information-theoretical setting, we obtain a scrambler which maps an arbitrary initial state to a distribution of quantum states that is close to Haar random in total variation distance. As a result, our PRSS exhibits a dispersing property. Loosely, it can span an $\epsilon$-net of the state space. This significantly strengthens what standard PRSGs can induce, as they may only concentrate on a small region of the state space as long as the average output state approximates a Haar random state in total variation distance. Our PRSS construction develops a parallel extension of the famous Kac's walk, and we show that it mixes exponentially faster than the standard Kac's walk. This constitutes the core of our proof. We also describe a few applications of PRSSs. While our PRSS construction assumes a post-quantum one-way function, PRSSs are potentially a weaker primitive and can be separated from one-way functions in a relativized world similar to standard PRSGs.

翻译：量子伪随机态生成器（PRSGs）近年来推动了令人兴奋的进展。PRSG在固定初始态（例如全零态）上输出一个在计算上不可与Haar随机态区分的输出态。然而，输出态的伪随机性在其他初始态上无法得到保证。事实上，已知的PRSG构造在某些初始态上被证明会失效。在本工作中，我们提出并构造了量子伪随机态搅乱器（PRSSs），它能在任意初始态上生成伪随机态。在信息论框架下，我们获得了一种搅乱器，能将任意初始态映射到总变差距离上接近Haar随机的量子态分布。因此，我们的PRSS展现出分散性质：粗略而言，它能张成状态空间的一个$\epsilon$-网。这显著强化了标准PRSGs所能实现的效果——后者可能仅集中于状态空间的某个小区域，只要平均输出态在总变差距离上近似于Haar随机态即可。我们的PRSS构造发展了著名Kac游走的并行扩展形式，并证明其混合速度远快于标准Kac游走——这构成了我们证明的核心。我们还描述了PRSSs的若干应用。尽管我们的PRSS构造假设存在后量子单向函数，但PRSSs可能是一种更弱的原始工具，并能在相对化世界中（类似于标准PRSGs）与单向函数相分离。