Pseudorandom unitaries are neither real nor sparse nor noise-robust

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.

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