We study the relationship between notions of pseudorandomness in the quantum and classical worlds. Pseudorandom quantum state generator (PRSG), a pseudorandomness notion in the quantum world, is an efficient circuit that produces states that are computationally indistinguishable from Haar random states. PRSGs have found applications in quantum gravity, quantum machine learning, quantum complexity theory, and quantum cryptography. Pseudorandom generators, on the other hand, a pseudorandomness notion in the classical world, is ubiquitous to theoretical computer science. While some separation results were known between PRSGs, for some parameter regimes, and PRGs, their relationship has not been completely understood. In this work, we show that a natural variant of pseudorandom generators called quantum pseudorandom generators (QPRGs) can be based on the existence of logarithmic output length PRSGs. Our result along with the previous separations gives a better picture regarding the relationship between the two notions. We also study the relationship between other notions, namely, pseudorandom function-like state generators and pseudorandom functions. We provide evidence that QPRGs can be as useful as PRGs by providing cryptographic applications of QPRGs such as commitments and encryption schemes. Our primary technical contribution is a method for pseudodeterministically extracting uniformly random strings from Haar-random states.

翻译：我们研究量子世界与经典世界中伪随机性概念之间的关系。量子伪随机态生成器（PRSG）是量子世界中的一种伪随机性概念，它是一种高效电路，能够产生与Haar随机态在计算上不可区分的量子态。PRSGs已在量子引力、量子机器学习、量子复杂性理论和量子密码学中得到应用。另一方面，经典伪随机生成器（PRG）是经典世界中的一种伪随机性概念，在理论计算机科学中无处不在。尽管在某些参数区间内，PRSGs与PRGs之间已知存在一些分离结果，但两者之间的关系尚未完全阐明。在本工作中，我们证明了一种称为量子伪随机生成器（QPRG）的PRG自然变体可以基于对数输出长度的PRSGs的存在性而构建。结合先前的分离结果，我们的结论为这两种概念之间的关系提供了更清晰的图景。我们还研究了其他概念之间的关系，即类伪随机函数态生成器与伪随机函数。通过展示QPRGs在承诺方案和加密方案等密码学应用中的实用性，我们提供了证据表明QPRGs可以像PRGs一样有用。我们的主要技术贡献在于提出了一种从Haar随机态中伪确定性提取均匀随机串的方法。