Quantum Relative Entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy (SQRE). We also introduce a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package DDS. In addition to handling QRE constraints, DDS accepts any combination of several other conic and non-conic convex constraints. Our comprehensive numerical experiments encompass several parts including 1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, 2) using DDS for combining QRE constraints with various other constraint types, and 3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols.

翻译：量子相对熵（QRE）规划是近年来备受关注且具有挑战性的凸优化问题类别，在量子计算与量子信息理论中具有重要应用。我们聚焦于基于QRE锥最优自协调障壁的现代内点（IP）方法。此类障壁函数与QRE锥相关的理论与数值难点阻碍了内点法的可扩展性。为应对这些挑战，我们提出一系列数值方法与线性代数技术及启发式策略，旨在提升自协调障壁函数的梯度与海森矩阵计算效率、优化线性系统求解及矩阵-向量乘积运算。同时介绍并探讨若干与QRE相关的概念，如对称量子相对熵（SQRE）。我们还提出一种用于执行面约简的两阶段方法，可显著提升QRE规划的性能。上述新技术已在软件包DDS的最新版本（DDS 2.2）中实现。除处理QRE约束外，DDS支持融合多种锥形与非锥形凸约束。综合数值实验涵盖以下部分：1）针对最近相关矩阵问题对DDS 2.2与Hypatia进行对比；2）利用DDS将QRE约束与其他多种约束类型相结合；3）计算量子密钥分发（QKD）信道的关键速率，并展示若干QKD协议的计算结果。