Quantum Speedups for Linear Programming via Interior Point Methods
Simon Apers, Sander Gribling · SIAM Journal on Computing · 2026
Abstract. We describe a quantum algorithm based on an interior point method for solving a linear program with [Formula: see text] inequality constraints on [Formula: see text] variables. The algorithm explicitly returns a feasible solution that is [Formula: see text]-close to optimal and runs in time [Formula: see text], which is sublinear for tall linear programs (i.e., [Formula: see text]). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [ Solving Linear Programs with Sqrt(rank) Linear System Solves, 2019]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the spectral approximation of [Formula: see text] for a tall matrix [Formula: see text]. The algorithm uses leverage score sampling in combination with Grover search and returns a [Formula: see text]-approximation by making [Formula: see text] row queries to [Formula: see text]. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [ SIAM J. Comput., 51 (2022), pp. 1703–1742]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi, and Jerbi [ Near-optimal quantum algorithms for multivariate mean estimation, 2022]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by preconditioning our random variable using our quantum algorithm for spectral approximation.