Pseudorandom quantum states (PRSs) and pseudorandom unitaries (PRUs) possess the dual nature of being efficiently constructible while appearing completely random to any efficient quantum algorithm. In this study, we establish fundamental bounds on pseudorandomness. We show that PRSs and PRUs exist only when the probability that an error occurs is negligible, ruling out their generation on noisy intermediate-scale and early fault-tolerant quantum computers. Additionally, we derive lower bounds on the imaginarity and coherence of PRSs and PRUs, rule out the existence of sparse or real PRUs, and show that PRUs are more difficult to generate than PRSs. Our work also establishes rigorous bounds on the efficiency of property testing, demonstrating the exponential complexity in distinguishing real quantum states from imaginary ones, in contrast to the efficient measurability of unitary imaginarity. Furthermore, we prove lower bounds on the testing of coherence. Lastly, we show that the transformation from a complex to a real model of quantum computation is inefficient, in contrast to the reverse process, which is efficient. Overall, our results establish fundamental limits on property testing and provide valuable insights into quantum pseudorandomness.

翻译：伪随机量子态（PRSs）和伪随机酉算子（PRUs）具有双重性质：既能高效构造，又对任何高效量子算法呈现出完全随机的特性。本研究建立了伪随机性的基本界限。我们证明，只有当错误发生概率可忽略时，PRSs和PRUs才存在，从而排除了在含噪中等规模和早期容错量子计算机上生成它们的可能性。此外，我们推导了PRSs和PRUs的虚部性和相干性的下界，排除了稀疏或实PRUs的存在性，并表明PRUs比PRSs更难生成。我们的工作还为属性测试的效率建立了严格界限，证明区分实量子态与虚量子态具有指数复杂性，这与酉算子虚部性的可高效测量性形成对比。进一步地，我们证明了相干性测试的下界。最后，我们表明从复数到实数模型的量子计算转换是低效的，而反向过程则是高效的。总体而言，我们的结果为属性测试建立了基本极限，并为量子伪随机性提供了重要见解。