Quantum mechanical framework for quantization-based optimization: from Gradient flow to Schroedinger equation

2026-03-12

AI Summary

This work addresses the challenge that quantized optimization algorithms often become trapped in local optima and lack global convergence guarantees in non-convex and combinatorial settings. It establishes, for the first time, a rigorous connection between such algorithms and quantum mechanics by modeling the sampling process as a gradient flow in a dissipative system. Through a transformation of the objective function, the authors derive a Schrödinger equation whose quantum tunneling effect enables efficient escape from local minima. By unifying perspectives from thermodynamics and quantum dynamics—integrating the Hamilton-Jacobi-Bellman and Fokker-Planck equations—the framework seamlessly bridges continuous and combinatorial optimization. Empirical results demonstrate significant performance gains over conventional methods on both non-convex continuous and combinatorial tasks, with successful applications to machine learning scenarios such as image classification.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• quantization-based optimization
• global convergence
• local minima escape
• combinatorial optimization
• nonconvex optimization

Innovation

• quantum tunneling
• gradient flow
• Schroedinger equation
• quantization-based optimization
• Fokker-Planck equation