Quantum relative entropy programs are convex optimization problems which minimize a linear functional 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. This has opened up the opportunity to use interior-point methods for nonsymmetric cone programs to solve these 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 programs 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 optimally self-concordant, even when these linear operators map to singular matrices. Second, we show how we can exploit a catalogue of common structures in these linear operators to compute the inverse Hessian products of the barrier function more efficiently. This step is typically the bottleneck when solving quantum relative entropy programs using interior-point methods, and therefore improving the efficiency of this step can significantly improve the computational performance of the algorithm. We demonstrate how these methods can be applied to important applications in quantum information theory, including quantum key distribution, quantum rate-distortion, quantum channel capacities, and estimating 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.

翻译：量子相对熵程序是一类凸优化问题，其目标是在量子相对熵函数的上境图仿射截面上最小化线性泛函。最近，针对该集合的自然障碍函数被证明具有自协调性。这为使用非对称锥规划的内点法求解此类优化问题开辟了新途径。本文展示了如何利用量子信息论应用中常见的结构特性，来提高使用内点法求解量子相对熵程序的效率。首先，我们证明由正线性算子复合的量子相对熵上境图的自然障碍函数具有最优自协调性，即使这些线性算子映射到奇异矩阵时依然成立。其次，我们展示了如何利用这些线性算子中的常见结构目录，来更高效地计算障碍函数的逆海森堡乘积。该步骤通常是使用内点法求解量子相对熵程序时的计算瓶颈，因此提升此步骤的效率能显著改善算法的计算性能。我们演示了这些方法如何应用于量子信息论中的重要领域，包括量子密钥分发、量子率失真理论、量子信道容量估计以及哈密顿量基态能量估算。数值结果表明，这些技术将计算时间提升了数个数量级，并使先前难以处理的问题得以求解。