Quantum mechanical framework for quantization-based optimization: from Gradient flow to Schroedinger equation
This work provides a foundational framework that unifies combinatorial and continuous optimization, extending to machine learning tasks like image classification, though it is incremental in its theoretical analysis.
The authors tackled the analysis of quantization-based optimization algorithms by modeling them as a gradient-flow dissipative system, which connects to the Schrödinger equation, revealing that quantum tunneling enables escape from local minima and guarantees global optimum access, with numerical experiments showing consistent outperformance over conventional algorithms.
This work presents a quantum mechanical framework for analyzing quantization-based optimization algorithms. The sampling process of the quantization-based search is modeled as a gradient-flow dissipative system, leading to a Hamilton-Jacobi-Bellman (HJB) representation. Through a suitable transformation of the objective function, this formulation yields the Schroedinger equation, which reveals that quantum tunneling enables escape from local minima and guarantees access to the global optimum. By establishing the connection to the Fokker-Planck equation, the framework provides a thermodynamic interpretation of global convergence. Such an analysis between the thermodynamic and the quantum dynamic methodology unifies combinatorial and continuous optimization, and extends naturally to machine learning tasks such as image classification. Numerical experiments demonstrate that quantization-based optimization consistently outperforms conventional algorithms across both combinatorial problems and nonconvex continuous functions.