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arXiv:2308.11594 · International Conference on Information and Communication Technology Convergence (ICTC)

Quantization-based Optimization with Perspective of Quantum Mechanics

Jinwuk Seok, Chang-Sik Cho

Published Oct 11, 2023• 0 citations
arXivHTMLPDF

Abstract

Statistical and stochastic analysis based on thermodynamics has been the main analysis framework for stochastic global optimization. Recently, with the appearance of quantum annealing or quantum tunneling algorithms for global optimization, we require a new research framework for global optimization algorithms. In this paper, we provide the analysis for quantization-based optimization based on the Schrödinger equation to reveal what property in quantum mechanics enables global optimization. We present that the tunneling effect derived by the Schrödinger equation in quantization-based optimization enables to escape of a local minimum. Additionally, we confirm that this tunneling effect is the same property included in quantum mechanics-based global optimization. Experiments with standard multi-modal benchmark functions represent that the proposed analysis is valid.

DOI
10.1109/ictc58733.2023.10393023

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Statistical and stochastic analysis based on thermodynamics has been the main analysis framework for stochastic global optimization. Recently, with the appearance of quantum annealing or quantum tunneling algorithms for global optimization, we require a new research framework for global optimization algorithms. In this paper, we provide the analysis for quantization-based optimization based on the Schrödinger equation to reveal what property in quantum mechanics enables global optimization. We present that the tunneling effect derived by the Schrödinger equation in quantization-based optimization enables to escape of a local minimum. Additionally, we confirm that this tunneling effect is the same property included in quantum mechanics-based global optimization. Experiments with standard multi-modal benchmark functions represent that the proposed analysis is valid.
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.
In this paper, we present an analysis of the tunneling effect in the quantization-based optimization. We prove that, when the quantization step size is an exponential decay function with respect to an iteration, the quantized objective function admits a decomposition into longitudinal and transverse fields. From the perspective of quantum mechanics, since the transverse field gives the quantum tunneling effect, quantization-based optimization can outperform a conventional thermodynamics-based optimization scheme. Experimental results for a vanilla objective function and a typical combinatorial optimization problem, such as the traveling salesman problem, demonstrate that the presented analysis is valid.
Quantum tunneling that helps particles escape from local minima has been applied in “quantum annealing” method to global optimization of nonlinear functions. To control size of kinetic energy of quantum particles, we form a “quantum tunneling parameter” QT≡m/HR2, where HR corresponds to a physical constant h, Planck's constant divided by 2π, that determines the lowest eigenvalue of quantum particles with mass m. Assumptions on profiles of the function V(x) around its minimum point x0, harmonic oscillator type and square well type, make us possible to write down analytical formulae of the kinetic energy K in terms of QT. The formulae tell that we can make quantum expectation value of particle coordinates x approximate to the minimum point x0 in QT→∞. For systems where we have almost degenerate eigenvalues, examination working with our QT, that x→x0 in QT→∞, is analytically shown also efficient. Similar results that x→x0 under QT→∞ are also obtained when we utilize random-walk quantum Monte Carlo method to represent tunneling phenomena according to conventional quantum annealing.
One of the greatest scientific achievements of physics in the 20th century is the discovery of quantum mechanics. The Schrodinger equation is the most fundamental equation in quantum mechanics describing the time-based evolution of the quantum state of a physical system. It has been found that the time-independent version of the equation can be derived from a general optimization algorithm. Instead of arguing for a new interpretation and possible deeper principle for quantum mechanics, this paper elaborates a few points of the equation as a general global optimization algorithm. Benchmarked against randomly generated hard optimization problems, this paper shows that the algorithm significantly outperformed a classic local optimization algorithm. The former found a solution in one second with a single trial better than the best one found by the latter around one hour after one hundred thousand trials.
Abstract We propose a number‐theory‐based quantized mathematical optimization scheme for various NP‐hard and similar problems. Conventional global optimization schemes, such as simulated and quantum annealing, assume stochastic properties that require multiple attempts. Although our quantization‐based optimization proposal also depends on stochastic features (i.e., the white‐noise hypothesis), it provides a more reliable optimization performance. Our numerical analysis equates quantization‐based optimization to quantum annealing, and its quantization property effectively provides global optimization by decreasing the measure of the level sets associated with the objective function. Consequently, the proposed combinatorial optimization method allows the removal of the acceptance probability used in conventional heuristic algorithms to provide a more effective optimization. Numerical experiments show that the proposed algorithm determines the global optimum in less operational time than conventional schemes.
Both reduction in computational load and setting parameters with ease are necessary in calculation of global optimization. For this we propose a new quantum mechanical method of the problem. Global optimization is achieved by a quantum particle with energy dissipation due to frictional force. Canonical quantization is applied to a system that consists of one mass point under friction proportional to its velocity. Realistic motion of a particle is abstracted from a resultant quantum system according to a causal interpretation. By simulation of the quantum particle motion under potential functions V(x) with local minima and one global minimum, it is clarified that the particle with its initial position at the local minimum at rest necessarily arrives at the global minimum. Only one particle is needed in the new algorithm. When we clarify mathematical structure of our algorithm, difficulties found in quantum annealing or conventional method for global optimization will thereby be solved in the new method.