Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra

2025-10-08

AI Summary

Precise spectral structure—eigenvalues and their degeneracies—of quantum Hamiltonians is essential for studying many-body physics and topological order, yet high-accuracy extraction is #BQP-complete. This work introduces QFAMES, a quantum algorithm that overcomes this complexity barrier under physically realistic assumptions. QFAMES integrates quantum filtering, phase estimation, and state preparation to achieve high-resolution identification of dominant eigenvalue clusters and exact degeneracy determination within a target energy window. Compared to conventional subspace methods, QFAMES offers provably improved sample complexity and superior resolution of degenerate eigenvalues, with rigorous theoretical guarantees. Numerical experiments demonstrate its high accuracy and robustness in characterizing quantum phase transitions in the transverse-field Ising model and estimating ground-state degeneracy in the 2D toric code.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Security and Privacy: Large-scale security measurements

Abstract

Fine-grained spectral properties of quantum Hamiltonians, including both eigenvalues and their multiplicities, provide useful information for characterizing many-body quantum systems as well as for understanding phenomena such as topological order. Extracting such information with small additive error is $# extsf{BQP}$-complete in the worst case. In this work, we introduce QFAMES (Quantum Filtering and Analysis of Multiplicities in Eigenvalue Spectra), a quantum algorithm that efficiently identifies clusters of closely spaced dominant eigenvalues and determines their multiplicities under physically motivated assumptions, which allows us to bypass worst-case complexity barriers. QFAMES also enables the estimation of observable expectation values within targeted energy clusters, providing a powerful tool for studying quantum phase transitions and other physical properties. We validate the effectiveness of QFAMES through numerical demonstrations, including its applications to characterizing quantum phases in the transverse-field Ising model and estimating the ground-state degeneracy of a topologically ordered phase in the two-dimensional toric code model. Our approach offers rigorous theoretical guarantees and significant advantages over existing subspace-based quantum spectral analysis methods, particularly in terms of the sample complexity and the ability to resolve degeneracies.

Problem

• Identifies clusters of dominant eigenvalues and their multiplicities
• Estimates observable expectation values within targeted energy clusters
• Bypasses worst-case complexity barriers under physical assumptions

Innovation

• Quantum algorithm identifies eigenvalue clusters and multiplicities
• Estimates observable values within targeted energy clusters
• Bypasses worst-case complexity with physical assumptions