🤖 AI Summary
Characterizing the existence boundaries of pseudorandom quantum states (PRS) and pseudorandom unitaries (PRU) under realistic constraints—namely, NISQ- and early-fault-tolerant hardware noise, real-valued quantum mechanics, and sparsity limitations.
Method: The authors introduce novel resource-theoretic measures—“pseudo-coherence” and “pseudo-imaginariness”—to rigorously quantify resource requirements across PRS, PRU, and pseudorandom shufflers (PRSS); they further analyze real–complex quantum model conversions and state distinguishability.
Contribution/Results: First proof that PRU fundamentally require complex amplitudes, whereas PRS admit constructions within purely real quantum mechanics; exponential lower bounds on the inefficiency of real-to-complex quantum model translation and on distinguishing real from imaginary quantum states; and demonstration that PRU are provably unconstructible on current quantum hardware—revealing intrinsic structural limitations of quantum pseudorandomness.
📝 Abstract
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. Further, we show that PRUs need imaginarity while PRS do not have this restriction. This implies that quantum randomness requires in general a complex-valued formalism of quantum mechanics, while for random quantum states real numbers suffice. Additionally, we derive lower bounds on the coherence of PRSs and PRUs, ruling out the existence of sparse PRUs and PRSs. We also show that the notions of PRS, PRUs and pseudorandom scramblers (PRSSs) are distinct in terms of resource requirements. We introduce the concept of pseudoresources, where states which contain a low amount of a given resource masquerade as high-resource states. We define pseudocoherence, pseudopurity and pseudoimaginarity, and identify three distinct types of pseudoresources in terms of their masquerading capabilities. 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. 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. Our results establish fundamental limits on property testing and provide valuable insights into quantum pseudorandomness.
