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Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications

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  • 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.

    Mathematics Subject Classification: Primary: 90C22, 90C30, 90C51, 81-08; Secondary: 90C25, 90C90.

    Citation:

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  • Table 2.  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
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    Table 1.  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
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