Quantum Pseudorandomness and Classical Complexity

TL;DR

This paper explores the relationship between quantum complexity classes and the existence of cryptographic pseudorandom quantum states, revealing nuanced distinctions and necessary assumptions in quantum cryptography and complexity theory.

Contribution

It constructs a quantum oracle where BQP equals QMA yet pseudorandom states exist, and proves that such states require certain complexity assumptions to exist.

Findings

Existence of a quantum oracle with BQP = QMA and pseudorandom states

Pseudorandom states do not exist if BQP = PP

Distinction between quantum and classical input algorithms

Abstract

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.