Finding quantum partial assignments by search-to-decision reductions
Jordi Weggemans
TL;DR
The paper addresses whether quantum witnesses admit search-to-decision reductions and shows a nuanced answer: while full quantum witnesses cannot, in general, be produced from a quantum oracle, one can efficiently obtain high-quality classical fingerprints in the form of density-matrix marginals for near-optimal witnesses when locality is constant. The authors introduce LEDMV, a $ obreak{ \\mathsf{QMA} }$-complete problem that blends the local Hamiltonian energy problem with local-density-consistency checks, and show how a circuit-to-Hamiltonian mapping (including a small-penalty clock) enables approximate reconstruction of near-optimal witnesses via a classical algorithm with access to a $ obreak{ \\mathsf{QMA} }$ oracle. They extend the approach to arbitrary problems in $ obreak{ \\mathsf{QMA} }$ by establishing approximately witness-preserving reductions, yielding density matrices whose entries approximate those of a witness with acceptance probability arbitrarily close to the optimum. This work advances understanding of quantum search-to-decision by showing a meaningful quantum analogue for partial information (density matrices) and leaves open the question of obtaining actual quantum states as witnesses and the impact of additional structural restrictions on the Hamiltonians involved.
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.