Quantum Pseudorandomness and Classical Complexity
This work addresses foundational questions in quantum complexity theory and cryptography, with implications for shadow tomography, but it is incremental in exploring nuances of existing assumptions.
The paper tackles the problem of understanding the relationship between quantum computational complexity classes and the existence of pseudorandom quantum states and unitaries, showing that under a quantum oracle, BQP equals QMA while pseudorandomness exists, but pseudorandom states do not exist if BQP equals PP.
We construct a quantum oracle relative to which $\mathsf{BQP} = \mathsf{QMA}$ but cryptographic pseudorandom quantum states and pseudorandom unitary transformations exist, a counterintuitive result in light of the fact that pseudorandom states can be "broken" by quantum Merlin-Arthur adversaries. We explain how this nuance arises as the result of a distinction between algorithms that operate on quantum and classical inputs. On the other hand, we show that some computational complexity assumption is needed to construct pseudorandom states, by proving that pseudorandom states do not exist if $\mathsf{BQP} = \mathsf{PP}$. We discuss implications of these results for cryptography, complexity theory, and shadow tomography.