Finding quantum partial assignments by search-to-decision reductions

2024-08-07

arXiv.org

AI Summary

Can the local reduced density matrices (e.g., $k$-body marginals) of an approximately optimal quantum witness for a $mathsf{QMA}$ problem be efficiently approximated using only classical polynomial-time access to a $mathsf{QMA}$ decision oracle—without preparing or manipulating the full quantum state? Method: We introduce the first adaptive classical algorithm that integrates a circuit-to-Hamiltonian mapping preserving near-optimal witnesses, local consistency checks, and density matrix reconstruction techniques. Contributions: (1) For any constant locality $k$ and inverse-polynomial precision, the $k$-body reduced density matrices of an approximately optimal $mathsf{QMA}$ witness can be computed in classical polynomial time via adaptive queries to a $mathsf{QMA}$ oracle; (2) we define a new $mathsf{QMA}$-complete problem, “Low-Energy Density Matrix Verification”; (3) we establish the first rigorous classical reduction framework from a $mathsf{QMA}$ decision oracle to extraction of local quantum information.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Local Search
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Explores quantum search-to-decision reductions in QMA.
• Constructs partial quantum assignments via density matrices.
• Develops classical algorithm for QMA witness approximations.

Innovation

• Quantum search-to-decision reduction
• Classical polynomial-time algorithm
• Low-energy Density Matrix Verification