State Synthesis with an NP oracle

Abstract

In this note, I describe a quantum algorithm that prepares an arbitrary state $|ψ\rangle$ to inverse exponential precision in polynomial time by making queries to an $\mathsf{NP}$ oracle, assuming one can directly query entries of the state vector corresponding to $|ψ\rangle$. The previous best upper bound on the complexity class required by an oracle for this task was $\mathsf{PP}$. I provide evidence that $\mathsf{NP}$ is the best possible upper bound by showing that any quantum algorithm using a $\mathsf{P}$ oracle, or even a $\mathsf{P/poly}$ oracle, requires exponentially many queries. I also discuss some implications of this result for the $\mathsf{QMA}$ search-to-decision problem and unitary synthesis.