Finding quantum partial assignments by search-to-decision reductions

Abstract: In computer science, many search problems are reducible to decision problems, which implies that finding a solution is as hard as deciding whether a solution exists. A quantum analogue of search-to-decision reductions would be to ask whether a quantum algorithm with access to a $\mathsf{QMA}$ oracle can construct $\mathsf{QMA}$ witnesses as quantum states. By a result from Irani, Natarajan, Nirkhe, Rao, and Yuen (CCC '22), it is known that this does not hold relative to a quantum oracle, unlike the cases of $\mathsf{NP}$, $\mathsf{MA}$, and $\mathsf{QCMA}$ where search-to-decision relativizes. We prove that if one is not interested in the quantum witness as a quantum state but only in terms of its partial assignments, i.e. the reduced density matrices, then there exists a classical polynomial-time algorithm with access to a $\mathsf{QMA}$ oracle that outputs approximations of the density matrices of a near-optimal quantum witness, for any desired constant locality and inverse polynomial error. Our construction is based on a circuit-to-Hamiltonian mapping that approximately preserves near-optimal $\mathsf{QMA}$ witnesses and a new $\mathsf{QMA}$-complete problem, Low-energy Density Matrix Verification, which is called by the $\mathsf{QMA}$ oracle to adaptively construct approximately consistent density matrices of a low-energy state.