Robust Interior Point Method for Quantum Key Distribution Rate Computation

Hao Hu1,2, Jiyoung Im1, Jie Lin3,4, Norbert Lütkenhaus3, and Henry Wolkowicz1

1Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
2Department of Mathematical Sciences, Clemson University, Clemson, SC, United States 29634
3Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
4Department of Electrical & Computer Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G4

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Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.

Quantum key distribution (QKD) is a provably secure key establishment protocol that can help safeguard important confidential information from the emerging threats of quantum computers. The core of a security proof of any QKD protocol is to calculate the secret key rate. This task that was once challenging even for experts has become more accessible to non-experts thanks to the development of numerical security proof methods. Specifically, the key rate calculation problem has been recast as a convex optimization problem. Consequently, many great tools developed by the convex optimization community can be applied to vastly extend the applicability and practicality of the proof methods. In this work, we apply and extend one of these tools, the facial reduction technique. We use this to reduce the dimension of the numerical problem size, and also improve stability by identifying and removing implicit redundant constraints and parts of the matrices of the objective function. The resulting algorithm is efficient and highly accurate. This allows us to solve previously intractable problems.

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