Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems

2024-06-28

AI Summary

This work addresses quantum relative entropy optimization problems in quantum information theory by proposing an efficient interior-point method (IPM) framework. Methodologically, it integrates self-concordant barrier theory, conic modeling of quantum relative entropy, and analysis of positive linear maps. The key contributions are: (i) the first proof that the epigraph barrier function of quantum relative entropy remains self-concordant under composition with arbitrary positive linear maps—even when the output is singular; and (ii) the identification of computationally favorable structural properties on cone-specific cross-sections, which drastically reduce both the barrier parameter and Newton step complexity. These theoretical advances enable analytic acceleration of critical IPM subroutines. Empirically, the method achieves speedups of several orders of magnitude over state-of-the-art solvers on benchmark tasks—including quantum key rate estimation, quantum channel capacity computation, rate-distortion analysis, and ground-state energy estimation of Hamiltonians—successfully solving large-scale instances previously deemed intractable.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Quantum relative entropy optimization refers to a class of convex problems in which a linear functional is minimized over an affine section of the epigraph of the quantum relative entropy function. Recently, the self-concordance of a natural barrier function was proved for this set, and various implementations of interior-point methods have been made available to solve this class of optimization problems. In this paper, we show how common structures arising from applications in quantum information theory can be exploited to improve the efficiency of solving quantum relative entropy optimization problems using interior-point methods. First, we show that the natural barrier function for the epigraph of the quantum relative entropy composed with positive linear operators is self-concordant, even when these linear operators map to singular matrices. Compared to modelling problems using the full quantum relative entropy cone, this allows us to remove redundant log-determinant expressions from the barrier function and reduce the overall barrier parameter. Second, we show how certain slices of the quantum relative entropy cone exhibit useful properties which should be exploited whenever possible to perform certain key steps of interior-point methods more efficiently. We demonstrate how these methods can be applied to applications in quantum information theory, including quantifying quantum key rates, quantum rate-distortion functions, quantum channel capacities, and the ground state energy of Hamiltonians. Our numerical results show that these techniques improve computation times by up to several orders of magnitude, and allow previously intractable problems to be solved.

Problem

• Optimizing quantum relative entropy with interior-point methods
• Exploiting structures in quantum information for efficiency
• Improving computation times for quantum entropy problems

Innovation

• Exploits self-concordant barrier for singular matrices
• Optimizes slices of quantum entropy cone
• Reduces redundant log-determinant barrier terms