We present a quantum interior point method with worst case running time $\widetilde{O}(\frac{n^{2.5}}{ξ^{2}} μκ^3 \log (1/ε))$ for SDPs and $\widetilde{O}(\frac{n^{1.5}}{ξ^{2}} μκ^3 \log (1/ε))$ for LPs, where the output of our algorithm is a pair of matrices $(S,Y)$ that are $ε$-optimal $ξ$-approximate SDP solutions. The factor $μ$ is at most $\sqrt{2}n$ for SDPs and $\sqrt{2n}$ for LP's, and $κ$ is an upper bound on the condition number of the intermediate solution matrices. For the case where the intermediate matrices for the interior point method are well conditioned, our method provides a polynomial speedup over the best known classical SDP solvers and interior point based LP solvers, which have a worst case running time of $O(n^{6})$ and $O(n^{3.5})$ respectively. Our results build upon recently developed techniques for quantum linear algebra and pave the way for the development of quantum algorithms for a variety of applications in optimization and machine learning.