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

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 [] using a quantum linear system algorithm []. The (worst case) work complexity of our method is, up to polylogarithmic factors, 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, an upper bound to the Frobenius norm of the linear systems of equations that appear, ‖M‖F, and 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 , or with exact methods, at least . 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.