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 (\epsilon)-optimality using (\mathcal{O} \left( \sqrt{m + n} \frac{R_{1}}{\epsilon}\right)) queries to an oracle that evaluates a potential function, where (R_{1}) is an (\ell_{1})-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 $(\mathcal{O} \left( \sqrt{m + n} \textsf{nnz} (A) \frac{R_1}{\epsilon}\right))( elementary gates, where )\textsf{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix.