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

Leonid Faybusovich; Cunlu Zhou

doi:10.3934/naco.2021017

Article Contents

2022, Volume 12, Issue 2: 445-467. Doi: 10.3934/naco.2021017

This issue Previous Article $ V $-$ E $-invexity in $ E $-differentiable multiobjective programming Next Article

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

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Abstract

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.

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

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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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