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June  2022, 12(2): 445-467. doi: 10.3934/naco.2021017

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

Leonid Faybusovich and  Cunlu Zhou

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

Received  June 2020 Revised  April 2021 Published  June 2022 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.

Keywords: long-step path-following algorithm, self-concordant functions, matrix monotone functions, quantum relative entropy, quantum key distribution, quantum information theory.
Mathematics Subject Classification: Primary: 90C22, 90C30, 90C51, 81-08; Secondary: 90C25, 90C90.
Citation: Leonid Faybusovich, Cunlu Zhou. Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 445-467. doi: 10.3934/naco.2021017
References:
[1]

F. Alizadeh, J. Haeberly and M. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM Journal on Optimization, 8 (1998), 746-768.  doi: 10.1137/S1052623496304700.

[2]

M. ApS, The MOSEK optimization toolbox for MATLAB manual, version 8.0 (revision 60), 2017, http://docs.mosek.com/8.0/toolbox/index.html.

[3]

C. Bachoc, D. C. Gijswijt, A. Schrijver and F. Vallentin, Invariant Semidefinite Programs, Springer US, Boston, MA, 2012. doi: 10.1007/978-1-4614-0769-0_9.

[4]

P. J. Coles, E. M. Metodiev and N. Ltkenhaus, Numerical approach for unstructured quantum key distribution, Nature Communications, 7 (2016), 11712.

[5]

B. Coutts, M. Girard and J. Watrous, Certifying optimality for convex quantum channel optimization problems, arXiv: 1810.13295, 2018.

[6]

D. den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming, Springer, Netherlands, 1994. doi: 10.1007/978-94-011-1134-8.

[7]

D. den Hertog, C. Roos and T. Terlaky, On the classical logarithmic barrier function method for a class of smooth convex programming problems, J. Optim. Theory Appl., 73 (1992), 1-25.  doi: 10.1007/BF00940075.

[8]

D. Drusvyatskiy and H. Wolkowicz, The Many Faces of Degeneracy in Conic Optimization, now, 2017.

[9]

H. Fawzi, J. Saunderson and P. A. Parrilo, Semidefinite approximations of the matrix logarithm, Found. Comput. Math., 19 (2019), 259-296.  doi: 10.1007/s10208-018-9385-0.

[10]

H. Fawzi and O. Fawzi, Efficient optimization of the quantum relative entropy, J. Phys. A. Math. Theory, 51 (2018), 154003. doi: 10.1088/1751-8121/aab285.

[11]

L. Faybusovich and C. Zhou, Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions, Comput. Optim. Appl., 72 (2019), 769-795.  doi: 10.1007/s10589-018-0054-7.

[12]

L. Faybusovich and C. Zhou, Self-concordance and matrix monotonicity with applications to quantum entanglement problems, Applied Mathematics and Computation, 375 (2020), 125071. doi: 10.1016/j.amc.2020.125071.

[13]

K. Fujisawa, M. Kojima and K. Nakata, Exploiting sparsity in primal-dual interior-point methods for semidefinite programming, Mathematical Programming, 79 (1997), 235-253.  doi: 10.1016/S0025-5610(97)00045-2.

[14] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511840371.
[15]

H.-K. Lo, M. Curty and K. Tamaki, Secure quantum key distribution, Nat. Photon., 8 (2014), 595-604. 

[16]

Y. Nesterov, Lectures on Convex Optimization, Springer International Publishing, 2018. doi: 10.1007/978-3-319-91578-4.

[17]

M. Pilanci and M. J. Wainwright, Newton sketch: A linear-time optimization algorithm with linear-quadratic convergence, 2015. doi: 10.1137/15M1021106.

[18]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys., 81 (2009), 1301-1350. 

[19]

K. C. Toh, M. J. Todd and R. H. Tütüncü, SDPT3 –- a MATLAB software package for semidefinite programming, optimization methods and software, Optimization Methods and Software, 11 (1999), 545-581.  doi: 10.1080/10556789908805762.

[20]

L. Vandenberghe and M. S. Andersen, Chordal Graphs and Semidefinite Optimization, Now Publishers, 2015.

[21]

A. Winick, N. Lütkenhaus and P. J. Coles, Reliable numerical key rates for quantum key distribution, Quantum, 2 (2018), 77.

show all references

References:
[1]

F. Alizadeh, J. Haeberly and M. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM Journal on Optimization, 8 (1998), 746-768.  doi: 10.1137/S1052623496304700.

[2]

M. ApS, The MOSEK optimization toolbox for MATLAB manual, version 8.0 (revision 60), 2017, http://docs.mosek.com/8.0/toolbox/index.html.

[3]

C. Bachoc, D. C. Gijswijt, A. Schrijver and F. Vallentin, Invariant Semidefinite Programs, Springer US, Boston, MA, 2012. doi: 10.1007/978-1-4614-0769-0_9.

[4]

P. J. Coles, E. M. Metodiev and N. Ltkenhaus, Numerical approach for unstructured quantum key distribution, Nature Communications, 7 (2016), 11712.

[5]

B. Coutts, M. Girard and J. Watrous, Certifying optimality for convex quantum channel optimization problems, arXiv: 1810.13295, 2018.

[6]

D. den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming, Springer, Netherlands, 1994. doi: 10.1007/978-94-011-1134-8.

[7]

D. den Hertog, C. Roos and T. Terlaky, On the classical logarithmic barrier function method for a class of smooth convex programming problems, J. Optim. Theory Appl., 73 (1992), 1-25.  doi: 10.1007/BF00940075.

[8]

D. Drusvyatskiy and H. Wolkowicz, The Many Faces of Degeneracy in Conic Optimization, now, 2017.

[9]

H. Fawzi, J. Saunderson and P. A. Parrilo, Semidefinite approximations of the matrix logarithm, Found. Comput. Math., 19 (2019), 259-296.  doi: 10.1007/s10208-018-9385-0.

[10]

H. Fawzi and O. Fawzi, Efficient optimization of the quantum relative entropy, J. Phys. A. Math. Theory, 51 (2018), 154003. doi: 10.1088/1751-8121/aab285.

[11]

L. Faybusovich and C. Zhou, Long-step path-following algorithm for solving symmetric programming problems with nonlinear objective functions, Comput. Optim. Appl., 72 (2019), 769-795.  doi: 10.1007/s10589-018-0054-7.

[12]

L. Faybusovich and C. Zhou, Self-concordance and matrix monotonicity with applications to quantum entanglement problems, Applied Mathematics and Computation, 375 (2020), 125071. doi: 10.1016/j.amc.2020.125071.

[13]

K. Fujisawa, M. Kojima and K. Nakata, Exploiting sparsity in primal-dual interior-point methods for semidefinite programming, Mathematical Programming, 79 (1997), 235-253.  doi: 10.1016/S0025-5610(97)00045-2.

[14] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.  doi: 10.1017/CBO9780511840371.
[15]

H.-K. Lo, M. Curty and K. Tamaki, Secure quantum key distribution, Nat. Photon., 8 (2014), 595-604. 

[16]

Y. Nesterov, Lectures on Convex Optimization, Springer International Publishing, 2018. doi: 10.1007/978-3-319-91578-4.

[17]

M. Pilanci and M. J. Wainwright, Newton sketch: A linear-time optimization algorithm with linear-quadratic convergence, 2015. doi: 10.1137/15M1021106.

[18]

V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus and M. Peev, The security of practical quantum key distribution, Rev. Mod. Phys., 81 (2009), 1301-1350. 

[19]

K. C. Toh, M. J. Todd and R. H. Tütüncü, SDPT3 –- a MATLAB software package for semidefinite programming, optimization methods and software, Optimization Methods and Software, 11 (1999), 545-581.  doi: 10.1080/10556789908805762.

[20]

L. Vandenberghe and M. S. Andersen, Chordal Graphs and Semidefinite Optimization, Now Publishers, 2015.

[21]

A. Winick, N. Lütkenhaus and P. J. Coles, Reliable numerical key rates for quantum key distribution, Quantum, 2 (2018), 77.

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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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
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
Table Options

Download as excel

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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