Classical and Quantum Speedups for Non-Convex Optimization via Energy Conserving Descent

We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation in a tractable model where the barrier-crossing mechanism can be computed explicitly. For one-dimensional double-well objectives in the under-guessing regime, we compute the expected dynamical hitting times from a local minimum to the global minimum. We prove that both sECD and qECD exhibit exponential improvements in continuous hitting time relative to their respective gradient-based baselines, stochastic gradient descent ({SGD}) and quantum tunneling walk (QTW). For objectives with tall barriers, qECD admits a further hitting time improvement over sECD. Mechanistically, ECD sidesteps the exponential cost associated with rare-escape events of {SGD} from local minima by moving from dissipative to energy-conserving dynamics.