Bibliographic Information: Ding, L., Hong, C., & Schilling, C. (2024). Ground and Excited States from Ensemble Variational Principles. Quantum.
Research Objective: This paper aims to address the limitations of the GOK variational principle in predicting the accuracy of individual eigenstates and eigenenergies when the ensemble energy is not fully converged. The authors aim to establish a quantitative relationship between the ensemble energy error and the errors in the individual states and energies.
Methodology: The authors utilize concepts from convex geometry, specifically Birkhoff polytopes and permutohedra, to analyze the GOK variational principle. They formulate the problem of minimizing and maximizing errors in the ensemble state, eigenstates, and eigenenergies as linear optimization problems over these geometric objects.
Key Findings: The paper proves that the errors in the ensemble state, individual eigenstates, and eigenenergies are linearly bounded by the ensemble energy error. This finding confirms the predictive power of the GOK variational principle even when the ensemble energy is not fully minimized. The authors derive explicit expressions for these bounds, which provide insights into the optimal choice of weights for practical applications.
Main Conclusions: The study validates the use of the GOK variational principle for targeting excited states in quantum systems, even in cases of imperfect convergence. The derived error bounds offer valuable guidance for designing and optimizing variational algorithms based on the GOK principle.
Significance: This research provides a rigorous mathematical foundation for the GOK variational principle, a cornerstone of many modern quantum computational methods. The findings have significant implications for fields like density functional theory, quantum Monte Carlo simulations, and variational quantum eigensolvers.
Limitations and Future Research: The study focuses on non-degenerate energy spectra. While the authors argue that the results can be generalized to degenerate cases, further investigation is needed to explore the implications of degeneracies in detail. Additionally, exploring the application of these findings to specific quantum algorithms and systems could provide valuable insights for practical implementations.
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by Lexin Ding, ... at arxiv.org 11-12-2024
https://arxiv.org/pdf/2401.12104.pdfDeeper Inquiries