Quantum Search-To-Decision Reductions and the State Synthesis Problem

It is a useful fact in classical computer science that many search problems are reducible to decision problems; this has led to decision problems being regarded as the $\textit{de facto}$ computational task to study in complexity theory. In this work, we explore search-to-decision reductions for quantum search problems, wherein a quantum algorithm makes queries to a classical decision oracle to output a desired quantum state. In particular, we focus on search-to-decision reductions for $\mathsf{QMA}$, and show that there exists a quantum polynomial-time algorithm that can generate a witness for a $\mathsf{QMA}$ problem up to inverse polynomial precision by making one query to a $\mathsf{PP}$ decision oracle. We complement this result by showing that $\mathsf{QMA}$-search does $\textit{not}$ reduce to $\mathsf{QMA}$-decision in polynomial-time, relative to a quantum oracle. We also explore the more general $\textit{state synthesis problem}$, in which the goal is to efficiently synthesize a target state by making queries to a classical oracle encoding the state. We prove that there exists a classical oracle with which any quantum state can be synthesized to inverse polynomial precision using only one oracle query and to inverse exponential precision using two oracle queries. This answers an open question of Aaronson from 2016, who presented a state synthesis algorithm that makes $O(n)$ queries to a classical oracle to prepare an $n$-qubit state, and asked if the query complexity could be made sublinear.