Interior Point Methods for Structured Quantum Relative Entropy Optimization Problems
Kerry He, James Saunderson, Hamza Fawzi
TL;DR
This work tackles quantum relative entropy optimization by introducing a self-concordant barrier for the epigraph of quasi-entropies composed with positive linear maps, even when maps target singular matrices. It then shows how problem structure—block-diagonal, low-rank, and entropy-difference forms—can be exploited to drastically accelerate interior-point methods, avoiding the inefficient lifting to the full quantum relative entropy cone. The authors develop tailored cone and barrier Oracles, implement them in software (QICS), and demonstrate up to several orders of magnitude speedups across quantum information tasks such as quantum key distribution, rate-distortion, channel capacities, and ground-state energy computations. The results reveal that exploiting structure yields substantial reductions in computation time and memory, enabling previously intractable high-dimensional problems to be solved with high accuracy using interior-point methods for nonsymmetric cones. Overall, the paper provides a practical toolkit for scalable quantum relative entropy optimization with broad implications for quantum information theory and related conic programming practice.
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.