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. Combining our approach with iterative refinement techniques, we obtain an exact solution to a linear optimization problem involving $m$ constraints and $n$ variables using at most $\mathcal{O} \left( (m + n) \text{nnz} (A) \kappa (\mathcal{M}) L \cdot \text{polylog} \left(m, n, \kappa (\mathcal{M}) \right) \right)$ elementary gates and $\mathcal{O} \left( \text{nnz} (A) L \right)$ classical arithmetic operations, where $ \text{nnz} (A)$ is the total number of non-zero elements found in the constraint matrix, $L$ denotes binary input length of the problem data, and $\kappa (\mathcal{M})$ is a condition number that depends only on the problem data.

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