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This work investigates the possibility of quantum speedups for continuous optimization through quantum Hamiltonian simulation. We establish the first rigorous query complexity bounds for unconstrained convex optimization via digital quantum annealing, based on the non-adiabatic Quantum Hamiltonian Descent (QHD) framework. In the process, we derive rigorous resource estimates for digital quantum simulation of Schrödinger operators depending only on input simulation parameters, given black-box evaluation access to separable, Lipschitz continuous potential b(t) f(x). We apply these simulation bounds to assess the complexity of high-dimensional convex optimization. Our annealing schedule achieves arbitrarily fast convergence rates in the evolution time, with computational time determined solely by the cost of discretization. We show that a G-Lipschitz convex function can be optimized to an error of ε with 𝒪̃(d^{1.5} G² R²/ε²) queries, given a starting point that is Euclidean distance R from optimal. Under reasonable assumptions such as the complexity of simulating Schrödinger operators, we show that Ω̃(d/ε²) queries are necessary. This suggests QHD does not offer improvements over classical methods in the noiseless zeroth order setting. However, we show that the QHD algorithm can tolerate 𝒪̃(ε³ /d^{1.5} G² R²) noise in function evaluation, and as a result, provides a super-quadratic query advantage over the best existing noise-tolerant classical algorithms in the high-dimensional setting. We leverage this to design a quantum algorithm for stochastic convex optimization that offers a super-quadratic speedup over all known classical and quantum algorithms in this regime. To our knowledge, these results represent the first rigorous quantum speedups for convex optimization obtained through a dynamical algorithm.