A quantum interior-point predictor-corrector algorithm for linear programming

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We introduce a new quantum optimization algorithm for dense linear programming problems, which can be seen as the quantization of the interior point predictor-corrector algorithm [1] using a quantum linear system algorithm [2]. The (worst case) work complexity of our method is, up to polylogarithmic factors, O(L root n(n+m)(parallel to M parallel to) over bar (F) (kappa)over bar> over bar epsilon(-2)) for n the number of variables in the cost function,mthe number of constraints,epsilon(-1)the target precision,Lthe bit length of the input data, (parallel to M parallel to) over bar (F) over bar <ian upper bound to the Frobenius norm of the linear systems of equations that appear, <(parallel to M parallel to)over bar>(F), and (kappa) over bar an upper bound to the condition number kappa of those systems of equations. This represents a quantum speed-up in the number n of variables in the cost function with respect to the comparable classical interior point algorithms when the initial matrix of the problem A is dense: if we substitute the quantum part of the algorithm by classical algorithms such as conjugate gradient descent, that would mean the whole algorithm has complexity O(L root n(n + m)(2) (kappa) over bar log(epsilon(-1))), or with exact methods, at least O(L root n(n + m)(2.373)). Also, in contrast with any quantum linear system algorithm, the algorithm described in this article outputs a classical description of the solution vector, and the value of the optimal solution.
© 2020 IOP Publishing Ltd. We acknowledge financial support from the Spanish MINECO Grants FIS2015-67411P, and the CAM research consortium QUITEMAD-CM, Grant No. S2018/TCS-4342. The research of MAM-D has been partially supported by the U.S. Army Research Office through Grant No. W911NF-14-1-0103. PAMC thanks the support of a FPU MECD Grant.
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