A quantum interior-point predictor–corrector algorithm for linear programming
P A M Casares, M A Martin-Delgado · Journal of Physics A Mathematical and Theoretical · 2020
Abstract 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 n ( n + m ) ‖ M ‖ F ¯ κ ¯ ϵ − 2 ) for n the number of variables in the cost function, m the number of constraints, ϵ −1 the target precision, L the bit length of the input data, ‖ M ‖ F ¯ an upper bound to the Frobenius norm of the linear systems of equations that appear, ‖ M ‖ F , and κ ¯ an upper bound to the condition number κ 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 n ( n + m ) 2 κ ¯ log ( ϵ − 1 ) ) , or with exact methods, at least O (