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Quantum Relative Entropy (QRE) programming is a recently popular andchallenging class of convex optimization problems with significant applicationsin quantum computing and quantum information theory. We are interested inmodern interior point (IP) methods based on optimal self-concordant barriersfor the QRE cone. A range of theoretical and numerical challenges associatedwith such barrier functions and the QRE cones have hindered the scalability ofIP methods. To address these challenges, we propose a series of numerical andlinear algebraic techniques and heuristics aimed at enhancing the efficiency ofgradient and Hessian computations for the self-concordant barrier function,solving linear systems, and performing matrix-vector products. We alsointroduce and deliberate about some interesting concepts related to QRE such assymmetric quantum relative entropy (SQRE). We also introduce a two-phase methodfor performing facial reduction that can significantly improve the performanceof QRE programming. Our new techniques have been implemented in the latestversion (DDS 2.2) of the software package DDS. In addition to handling QREconstraints, DDS accepts any combination of several other conic and non-conicconvex constraints. Our comprehensive numerical experiments encompass severalparts including 1) a comparison of DDS 2.2 with Hypatia for the nearestcorrelation matrix problem, 2) using DDS for combining QRE constraints withvarious other constraint types, and 3) calculating the key rate for quantum keydistribution (QKD) channels and presenting results for several QKD protocols.

Efficient Implementation of Interior-Point Methods for Quantum Relative Entropy