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

Abstract: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.

Quantum Physics (quant-ph); Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

arXiv:2604.13022 [quant-ph]

(or arXiv:2604.13022v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2604.13022 Focus to learn more arXiv-issued DOI via DataCite

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