A quantum central path algorithm for linear optimization

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a single simulation working directly with the nonlinear complementarity equations. This approach yields an algorithm for solving linear optimization problems involving m constraints and n variables to varepsilon-optimality using O left( √m + n fracR1varepsilonright) queries to an oracle that evaluates a potential function, where R1 is an ℓ1-norm upper bound on the size of the optimal solution. In the standard gate model (i.e., without access to quantum RAM) our algorithm can obtain highly-precise solutions to LO problems using at most O left( √m + n textsfnnz (A) fracR1varepsilonright) elementary gates, where textsfnnz (A) is the total number of non-zero elements found in the constraint matrix.