Researchers introduce a new theoretical tool and use it to show that several proposed ways to make quantum pseudorandom objects would actually create hard classical puzzles. At issue are pseudorandom states (quantum versions of random-looking quantum states) and pseudorandom unitaries (quantum operations that look random). A one-way function is a simple kind of mathematical object that is easy to compute but hard to reverse; such functions are a basic building block in classical cryptography. The paper shows that many recent quantum proposals that were intended to avoid building on one-way functions instead imply those functions or hard problems in NP (a standard class of computationally difficult problems).

The main technical advance is a procedure the authors call NP-aided shadow tomography. Ordinary shadow tomography is a way to learn useful features of an unknown quantum state by making many measurements. NP-aided shadow tomography lets an efficient classical algorithm learn collections of what the authors call "computable pure states" when the algorithm is allowed help from an NP witness. A computable pure state, in plain terms, is a quantum state whose amplitudes and phases on any basis term can be calculated efficiently on a classical computer if you know an efficient recipe that prepares the state.

Using this idea, the authors prove that collections of such computable pure states can be learned efficiently with NP help. They also give algorithms that learn quantum operations (unitaries) when the learner can make a polynomial number of queries to the operation and use NP-style witnesses. "Polynomially many" here means the number of queries grows reasonably with the size of the input, not exponentially.

Putting these ingredients together, the paper analyzes several existing constructions for pseudorandom states and pseudorandom unitaries. The authors find that a number of architectures, including a proposal called Hamiltonian Phase States (cited as Bostanci et al., TQC 2025), would yield classical one-way functions or would imply that NP-hard problems can be solved. In other words, if those quantum constructions work as intended, they would also give strong classical complexity consequences that many researchers had hoped to avoid.