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doi: 10.3934/naco.2021017
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## Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications

 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA

Received  June 2020 Revised  April 2021 Early access May 2021

We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult optimization problem involving the quantum relative entropy in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.

Citation: Leonid Faybusovich, Cunlu Zhou. Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021017
##### References:

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##### References:
Numerical results for QKD optimization problem (68)
 Long-Step Path-Following cvxquad $+$ mosek $n$ $k$ $m$ $r_1$ $r_2$ $T_{ac}$(s) $T_{pf}$(s) $nNewton$ $f_{min}$ Time(s) $f_{min}$ 4 8 2 2 2 0.15 0.03 6 0.2744 40.39 0.2744 6 12 4 1 2 0.15 0.15 14 0.0498 2751.39 0.0498 12 24 6 2 4 0.17 0.75 13 0.0440 N/A failed 16 32 10 2 2 0.19 1.69 10 0.0511 N/A failed 32 64 20 2 2 0.61 54.34 10 0.0332 N/A failed
 Long-Step Path-Following cvxquad $+$ mosek $n$ $k$ $m$ $r_1$ $r_2$ $T_{ac}$(s) $T_{pf}$(s) $nNewton$ $f_{min}$ Time(s) $f_{min}$ 4 8 2 2 2 0.15 0.03 6 0.2744 40.39 0.2744 6 12 4 1 2 0.15 0.15 14 0.0498 2751.39 0.0498 12 24 6 2 4 0.17 0.75 13 0.0440 N/A failed 16 32 10 2 2 0.19 1.69 10 0.0511 N/A failed 32 64 20 2 2 0.61 54.34 10 0.0332 N/A failed
Numerical Results for (63)
 long-step path-following $n$ $m$ $N$ $f_{min}$ $nNewton$ $T_{ac}$(s) $T_{pf}$(s) 4 2 4 27.3538 7 0.18 0.01 8 4 8 8.3264 13 0.19 0.03 16 8 16 18.4274 13 0.26 0.09 32 16 32 39.2516 21 1.39 1.06 64 32 64 91.6534 27 26.34 47.25
 long-step path-following $n$ $m$ $N$ $f_{min}$ $nNewton$ $T_{ac}$(s) $T_{pf}$(s) 4 2 4 27.3538 7 0.18 0.01 8 4 8 8.3264 13 0.19 0.03 16 8 16 18.4274 13 0.26 0.09 32 16 32 39.2516 21 1.39 1.06 64 32 64 91.6534 27 26.34 47.25
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