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）规划是一类近期备受关注且具有挑战性的凸优化问题，在量子计算与量子信息理论中具有重要应用。我们聚焦于基于量子相对熵锥最优自和谐障碍函数的现代内点（IP）方法。此类障碍函数与量子相对熵锥面临的一系列理论与数值挑战，制约了内点法的可扩展性。为解决这些挑战，我们提出了一系列数值与线性代数技术及启发式策略，旨在提升自和谐障碍函数梯度与海森矩阵的计算效率、求解线性系统以及执行矩阵-向量乘积。我们还介绍并探讨了与量子相对熵相关的一些有趣概念，如对称量子相对熵（SQRE）。此外，我们提出了一种用于执行面约简的两阶段方法，该方法可显著提升量子相对熵规划的性能。这些新技术已集成至最新版软件包DDS（2.2版）中。除处理量子相对熵约束外，DDS还能兼容其他多种锥形与非锥形凸约束的任意组合。我们的综合数值实验涵盖多个部分，包括：1）将DDS 2.2与Hypatia在最近相关矩阵问题上进行对比；2）利用DDS将QRE约束与其他多种约束类型进行联合求解；3）计算量子密钥分发（QKD）信道的关键速率，并展示多种QKD协议的实验结果。