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）规划是近期广受关注且具有挑战性的一类凸优化问题，在量子计算与量子信息理论中具有重要应用。本文研究基于QRE锥最优自协调障碍函数的现代内点（IP）方法。与此类障碍函数及QRE锥相关的诸多理论与数值挑战，长期以来制约着内点法的可扩展性。为应对这些挑战，我们提出了一系列数值与线性代数技术及启发式方法，旨在提升自协调障碍函数的梯度与Hessian矩阵计算、线性方程组求解以及矩阵-向量乘积运算的效率。同时，我们引入并探讨了与QRE相关的若干重要概念，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，可显著提升QRE规划的性能。这些新技术已集成至软件包DDS的最新版本（DDS 2.2）中。除处理QRE约束外，DDS可同时接受多种其他锥形与非锥形凸约束的任意组合。我们通过系统的数值实验开展多维度评估：1）针对最近相关矩阵问题，对比DDS 2.2与Hypatia的性能；2）利用DDS实现QRE约束与其他类型约束的联合处理；3）计算量子密钥分发（QKD）信道的密钥率，并对多种QKD协议给出计算结果。