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

Jinwuk Seok, Changsik Cho

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.

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

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.
Paper Structure (43 sections, 15 theorems, 107 equations, 8 figures, 8 tables, 3 algorithms)

This paper contains 43 sections, 15 theorems, 107 equations, 8 figures, 8 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Non-convex optimization based on quantum mechanics
  4. Non-convex optimization based on thermodynamics
  5. Preliminaries
  6. Definition and Assumption
  7. Fundamental Process of the Quantization-based Search from the Perspective of Level sets
  8. Dynamic analysis of the search process
  9. Hamiltonian based Analysis
  10. Quantum Mechanical and Thermodynamical Analysis
  11. Experimental Results
  12. Combinatorial and Non-Convex Optimization
  13. Machine Learning
  14. Conclusion
  15. Appendix
  16. ...and 28 more sections

Key Result

Theorem 3.1

The derivative of $V^Q$ with respect to $t$ tends to zero, i.e., $\partial_t V^Q(\boldsymbol{x}_t, t) \rightarrow 0$, as the quantization step size decreases to zero with increasing $t$, that is, as $Q_p^{-1}(t) \rightarrow 0$.

Figures (8)

  • Figure 1: Time indices $\tau$, $t$, and $\bar{t}$. $\tau$ is the basic index, as defined in Algorithm \ref{['alg-1']}. $t$ denotes the time index for $f_{\text{opt}}^Q$, which is updated when $f^Q \leq f_{\text{opt}}^Q$. $\bar{t}$ is updated whenever $f^Q < f_{\text{opt}}^Q$. The red line indicates the trend of $f_{\tau}^Q$.
  • Figure 2: Convergence trends for each algorithm in the 125-city TSP experiment. The quantization-based optimization scheme exhibits a faster convergence property compared to other algorithms.
  • Figure 3: Illustrative diagram of virtual functions used for analysis. Left: Linear virtual function — at $\boldsymbol{x}_t$, the gradient of the real objective function matches that of the virtual function. Right: Sinusoidal virtual function — at $\boldsymbol{x}_t$, the gradient of the real objective function again matches that of the virtual function.
  • Figure 4: Conceptual diagram of the tunneling effect. Left: in the equality case, Algorithm \ref{['alg-1']} tunnels the state vector through the energy barrier at the same quantized objective value. Right: in the inequality case, Algorithm \ref{['alg-1']} tunnels the state vector through the barrier at a lower quantized objective value.
  • Figure 5: Comparison of 100-city TSP routes provided by each optimization algorithm
  • ...and 3 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma C.1
  • Lemma C.2
  • Theorem C.1
  • ...and 9 more