Long-Step Path-Following Algorithm for Quantum Information Theory: Some Numerical Aspects and Applications

TL;DR

This work develops and analyzes a long-step path-following algorithm for a broad class of convex problems with nonlinear objectives and semidefinite constraints, with a focus on quantum information theory. It derives analytic Hessians and vectorized implementations, and demonstrates two specialized problem types (Type I and II) as well as matrix-monotone objective structures, including detailed gradient/Hessian formulas. The approach is empirically validated on challenging quantum information tasks, notably a quantum key distribution optimization based on quantum relative entropy, where it achieves substantial speedups (up to thousands of times faster) over existing first-order methods and competitors. The results indicate that many quantum-information optimization problems are compatible with standard self-concordant barriers, enabling robust, accurate, and scalable second-order methods; the authors also outline future work on theoretical guarantees, barrier generalization, and scalability enhancements via sparsity and sketching.

Abstract

We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult and important optimization problem in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Remark 2.6
• Theorem 2.7
• Theorem 2.8
• Theorem 2.9
• Theorem 3.1