On the Limitations of Pseudorandom Unitaries

Pseudorandom unitaries (PRUs), one of the key quantum pseudorandom notions, are efficiently computable unitaries that are computationally indistinguishable from Haar random unitaries. While there is evidence to believe that PRUs are weaker than one-way functions, so far its relationship with other quantum cryptographic primitives (that are plausibly weaker than one-way functions) has not been fully established. In this work, we focus on quantum cryptographic primitives with classical communication, referred to as QCCC primitives. Our main result shows that QCCC bit commitments and QCCC key agreement, cannot be constructed from pseudorandom unitaries in a black-box manner. Our core technical contribution is to show (in a variety of settings) the difficulty of distinguishing identical versus independent Haar unitaries by separable channels. Our result strictly improves upon prior works which studied similar problems in the context of learning theory [Anshu, Landau, Liu, STOC 2022] and cryptography [Ananth, Gulati, Lin, TCC 2024].