Optimization algorithm entropy based on quantum dynamics

doi:10.11772/j.issn.1001-9081.2023121760

Journal of Computer Applications ›› 2025, Vol. 45 ›› Issue (1): 186-195.DOI: 10.11772/j.issn.1001-9081.2023121760

• Advanced computing • Previous Articles Next Articles

Optimization algorithm entropy based on quantum dynamics

Quan TANG¹^,²^,³, Peng WANG⁴(), Gang XIN⁴

^1.Chengdu Institution of Computer Application，China Academy of Science，Chengdu Sichuan 610213，China
^2.University of Chinese Academy of Sciences，Beijing 100049，China
^3.Chengdu Jincheng College，School of Electronic Information，Chengdu Sichuan，611731，China
^4.School of Computer and Artificial Intelligence，Southwest Minzu University，Chengdu Sichuan 610225，China

Received:2023-12-19 Revised:2024-03-20 Accepted:2024-04-10 Online:2024-05-07 Published:2025-01-10
Contact: Peng WANG
About author:TANG Quan， born in 1981， Ph. D. candidate. Her research interests include quantum inspired algorithm， swarm intelligence algorithm.
XIN Gang， born in 1983， Ph. D.， lecturer. His research interests include quantum inspired algorithm， high-performance computing.
Supported by:
Fundamental Research Funds for the Central Universities - Southwest Minzu University(2020NYB18)

基于量子动力学的优化算法熵

唐泉¹^,²^,³, 王鹏⁴(), 辛罡⁴

^1.中国科学院成都计算机应用研究所，成都 610213
^2.中国科学院大学，北京 100049
^3.成都锦城学院电子信息学院，成都 611731
^4.西南民族大学计算机与人工智能学院，成都 610225

通讯作者: 王鹏
作者简介:唐泉（1981—），女，四川营山人，博士研究生，主要研究方向：量子启发式算法、群智能算法；
辛罡（1983—），男，四川成都人，讲师，博士，CCF会员，主要研究方向：量子启发式算法、高性能计算。
基金资助:
西南民族大学中央高校基本科研业务费专项资金资助项目(2020NYB18)

Abstract

Abstract:

Entropy is a common description method in the analysis and research of optimization system. To address the lack of in-depth analysis of the inherent relationship between the dynamic behavior and entropy of different optimization systems， an optimization algorithm entropy based on quantum dynamics was proposed. Firstly， based on the similarity between Brownian motion and sampling behavior in physics， a Brownian motion description method for optimization problems was proposed. The mechanical expression of optimization problems was transformed into the form of energy and introduced into the Schr?dinger equation， and an optimization algorithm based on quantum dynamics was proposed. Then， the probability expression of optimization problems under the Schr?dinger equation was combined to obtain optimization algorithm entropy. Finally， the random behavior of particles under constraint of the objective function was analyzed， and the relationship between basic search behavior of optimization systems under quantum dynamics and entropy was given. By tracking and analyzing the dynamic behavior and entropy change trend of optimization systems from three different aspects： reference energy， free particle kinetic energy， and objective function disturbance， the correlation between entropy and search behavior of optimization systems was verified through experiments. Experiments results show that optimization algorithm entropy based on quantum dynamics can deeply analyze optimization process， providing a new idea and method for studying optimization algorithms.

Key words: quantum dynamics, optimization problem, Brownian motion, Schr?dinger equation, entropy

摘要：

在优化系统分析和研究中，熵是一种常用的描述手段，针对不同优化系统动态行为和熵之间的内在联系缺乏深入分析的问题，提出一种基于量子动力学的优化算法熵。首先基于物理学中的布朗运动与采样行为的相似性提出优化问题的布朗运动描述方法。将优化问题力学表达转化为能量的形式引入薛定谔方程，提出基于量子动力学的优化算法；然后结合优化问题在薛定谔方程下的概率表达得到优化算法熵；最后对目标函数约束下的粒子随机行为进行分析，给出了量子动力学下优化系统的基本搜索行为与熵的关系。实验从参考能量、自由粒子动能和目标函数扰动3个不同方面跟踪和分析优化系统的动态行为和熵的变化趋势，验证了熵与优化系统搜索行为之间的相关性。实验结果表明，基于量子动力学的优化算法熵可以深入分析优化过程，为研究优化算法给出了新的思路和方法。

关键词: 量子动力学, 优化问题, 布朗运动, 薛定谔方程, 熵

CLC Number:

TP301.6

Quan TANG, Peng WANG, Gang XIN. Optimization algorithm entropy based on quantum dynamics[J]. Journal of Computer Applications, 2025, 45(1): 186-195.

唐泉, 王鹏, 辛罡. 基于量子动力学的优化算法熵[J]. 《计算机应用》唯一官方网站, 2025, 45(1): 186-195.

Figures/Tables 8

Fig. 1 Schematic diagram of sin function superimposed with weak random interference

Fig. 2 Schematic diagram of sin function superimposed with strong random interference

Tab. 1 Test functions

函数	名称	表达式	定义域
f₁	Girewank	$f 1 (x) = x i 2 4 000 - ∏ i = 1 n c o s x i i + 1$	［-100，100］
f₂	Ackley	$f 2 (x) = - a e x p - b 1 n ∑ i = 1 n x i 2 - e x p 1 n ∑ i = 1 n c o s (c x i) + a + e x p (1)$	［-32.7，32.7］
f₃	Levy	$f 3 (x) = s i n 2 (π ω 1) + ∑ i = 1 n - 1 (ω i - 1) 2 [1 + 10 s i n 2 (π ω i + 1)] + (ω n - 1) 2 [1 + s i n 2 (2 π ω n)]; ω i = 1 + x i - 1 4, i = 1,2, …, n$	［-10，10］
f₄	Rastrigin	$f 4 (x) = 10 n + ∑ i = 1 n [x i 2 - 10 c o s (2 π x i)]$	［-5.12，5.12］
f₅	Sphere	$f 5 (x) = ∑ i = 1 n x i 2$	［-5.12，5.12］
f₆	Sum squares	$f 6 (x) = ∑ i = 1 n i x i 2$	［-10，10］
f₇	Zakharov	$f 7 (x) = ∑ i = 1 n x i 2 + ∑ i = 1 n 0.5 i x i 2 + ∑ i = 1 n 0.5 i x i 4$	［5，10］

Tab. 1 Test functions

函数	名称	表达式	定义域
f₁	Girewank	$f 1 (x) = x i 2 4 000 - ∏ i = 1 n c o s x i i + 1$	［-100，100］
f₂	Ackley	$f 2 (x) = - a e x p - b 1 n ∑ i = 1 n x i 2 - e x p 1 n ∑ i = 1 n c o s (c x i) + a + e x p (1)$	［-32.7，32.7］
f₃	Levy	$f 3 (x) = s i n 2 (π ω 1) + ∑ i = 1 n - 1 (ω i - 1) 2 [1 + 10 s i n 2 (π ω i + 1)] + (ω n - 1) 2 [1 + s i n 2 (2 π ω n)]; ω i = 1 + x i - 1 4, i = 1,2, …, n$	［-10，10］
f₄	Rastrigin	$f 4 (x) = 10 n + ∑ i = 1 n [x i 2 - 10 c o s (2 π x i)]$	［-5.12，5.12］
f₅	Sphere	$f 5 (x) = ∑ i = 1 n x i 2$	［-5.12，5.12］
f₆	Sum squares	$f 6 (x) = ∑ i = 1 n i x i 2$	［-10，10］
f₇	Zakharov	$f 7 (x) = ∑ i = 1 n x i 2 + ∑ i = 1 n 0.5 i x i 2 + ∑ i = 1 n 0.5 i x i 4$	［5，10］

Fig. 3 Comparison experiment results of algorithm entropy and energy under different reference energy

Fig. 4 Algorithm entropy variances under different reference energies （ER）

Tab. 2 Experimental results of algorithm entropy and energy under different kinetic energy of free particles

函数	函数类型	物理量	统计指标	$σ i n t$	$σ i n t$ /2	$σ i n t$ /4	$σ i n t$ /8	$σ i n t$ /16	$σ i n t$ /32	$σ i n t$ /64
$f 1$	多模	熵	Mean	6.928	6.922	6.860	6.276	5.576	5.023	3.890
			Std	0.004	0.001	0.009	0.133	0.248	0.116	0.000
			Max	6.931	6.923	6.866	6.432	5.915	5.170	3.890
			Min	6.922	6.921	6.853	6.097	5.127	4.745	3.890
		能量	Mean	18.126	17.639	16.120	12.806	10.384	8.208	3.655
			Std	0.293	0.026	0.132	0.481	0.553	0.230	0.000
			Max	18.276	17.659	16.210	13.432	11.168	8.679	3.655
			Min	17.663	17.618	16.030	12.191	9.408	7.782	3.655
$f 2$	多模	熵	Mean	8.761	8.754	8.671	8.280	7.563	6.925	6.244
			Std	0.004	0.001	0.007	0.045	0.098	0.097	0.109
			Max	8.763	8.755	8.675	8.324	7.696	7.057	6.372
			Min	8.754	8.753	8.666	8.231	7.432	6.792	6.065
		能量	Mean	18.165	17.938	16.979	14.668	11.170	7.851	5.199
			Std	0.128	0.011	0.067	0.214	0.361	0.387	0.271
			Max	18.229	17.947	17.023	14.906	11.677	8.378	5.554
			Min	17.955	17.929	16.934	14.434	10.704	7.315	4.747
$f 3$	多模	熵	Mean	7.598	7.590	7.507	7.170	6.496	5.623	5.050
			Std	0.005	0.001	0.008	0.045	0.104	0.067	0.078
			Max	7.600	7.592	7.512	7.210	6.638	5.688	5.110
			Min	7.590	7.589	7.502	7.108	6.270	5.576	4.988
		能量	Mean	3.550	3.318	2.564	1.568	0.804	0.305	0.118
			Std	0.130	0.028	0.050	0.088	0.085	0.017	0.018
			Max	3.623	3.343	2.600	1.663	0.933	0.328	0.133
			Min	3.331	3.293	2.530	1.449	0.633	0.294	0.102
$f 5$	单模	熵	Mean	6.928	6.921	6.833	6.422	5.787	5.114	4.415
			Std	0.005	0.001	0.008	0.041	0.089	0.125	0.095
			Max	6.931	6.922	6.839	6.451	5.887	5.262	4.530
			Min	6.921	6.920	6.827	6.386	5.670	4.905	4.263
		能量	Mean	8.312	7.744	5.682	2.576	0.916	0.268	0.061
			Std	0.318	0.023	0.131	0.210	0.148	0.064	0.012
			Max	8.470	7.765	5.781	2.748	1.099	0.356	0.077
			Min	7.787	7.722	5.588	2.386	0.712	0.168	0.042
$f 6$	单模	熵	Mean	7.598	7.591	7.509	7.128	6.530	5.754	5.049
			Std	0.004	0.001	0.008	0.038	0.075	0.105	0.096
			Max	7.600	7.592	7.515	7.157	6.619	5.887	5.175
			Min	7.590	7.589	7.502	7.093	6.416	5.587	4.906
		能量	Mean	31.759	29.563	21.928	10.136	3.547	0.968	0.230
			Std	1.164	0.093	0.526	0.856	0.621	0.216	0.045
			Max	32.314	29.652	22.353	10.833	4.333	1.266	0.285
			Min	29.717	29.478	21.455	9.333	2.667	0.636	0.162
$f 7$	单模	熵	Mean	7.309	7.295	7.179	6.822	6.137	5.361	4.657
			Std	0.006	0.003	0.012	0.029	0.118	0.119	0.127
			Max	7.312	7.298	7.187	6.850	6.292	5.516	4.822
			Min	7.299	7.291	7.168	6.791	5.976	5.211	4.479
		能量	Mean	107.060	87.859	45.883	13.991	3.197	0.614	0.146
			Std	9.016	2.787	2.725	1.483	0.710	0.117	0.029
			Max	113.224	90.532	47.785	15.502	4.226	0.773	0.182
			Min	92.088	85.323	43.586	12.346	2.205	0.464	0.102

Tab. 2 Experimental results of algorithm entropy and energy under different kinetic energy of free particles

函数	函数类型	物理量	统计指标	$σ i n t$	$σ i n t$ /2	$σ i n t$ /4	$σ i n t$ /8	$σ i n t$ /16	$σ i n t$ /32	$σ i n t$ /64
$f 1$	多模	熵	Mean	6.928	6.922	6.860	6.276	5.576	5.023	3.890
			Std	0.004	0.001	0.009	0.133	0.248	0.116	0.000
			Max	6.931	6.923	6.866	6.432	5.915	5.170	3.890
			Min	6.922	6.921	6.853	6.097	5.127	4.745	3.890
		能量	Mean	18.126	17.639	16.120	12.806	10.384	8.208	3.655
			Std	0.293	0.026	0.132	0.481	0.553	0.230	0.000
			Max	18.276	17.659	16.210	13.432	11.168	8.679	3.655
			Min	17.663	17.618	16.030	12.191	9.408	7.782	3.655
$f 2$	多模	熵	Mean	8.761	8.754	8.671	8.280	7.563	6.925	6.244
			Std	0.004	0.001	0.007	0.045	0.098	0.097	0.109
			Max	8.763	8.755	8.675	8.324	7.696	7.057	6.372
			Min	8.754	8.753	8.666	8.231	7.432	6.792	6.065
		能量	Mean	18.165	17.938	16.979	14.668	11.170	7.851	5.199
			Std	0.128	0.011	0.067	0.214	0.361	0.387	0.271
			Max	18.229	17.947	17.023	14.906	11.677	8.378	5.554
			Min	17.955	17.929	16.934	14.434	10.704	7.315	4.747
$f 3$	多模	熵	Mean	7.598	7.590	7.507	7.170	6.496	5.623	5.050
			Std	0.005	0.001	0.008	0.045	0.104	0.067	0.078
			Max	7.600	7.592	7.512	7.210	6.638	5.688	5.110
			Min	7.590	7.589	7.502	7.108	6.270	5.576	4.988
		能量	Mean	3.550	3.318	2.564	1.568	0.804	0.305	0.118
			Std	0.130	0.028	0.050	0.088	0.085	0.017	0.018
			Max	3.623	3.343	2.600	1.663	0.933	0.328	0.133
			Min	3.331	3.293	2.530	1.449	0.633	0.294	0.102
$f 5$	单模	熵	Mean	6.928	6.921	6.833	6.422	5.787	5.114	4.415
			Std	0.005	0.001	0.008	0.041	0.089	0.125	0.095
			Max	6.931	6.922	6.839	6.451	5.887	5.262	4.530
			Min	6.921	6.920	6.827	6.386	5.670	4.905	4.263
		能量	Mean	8.312	7.744	5.682	2.576	0.916	0.268	0.061
			Std	0.318	0.023	0.131	0.210	0.148	0.064	0.012
			Max	8.470	7.765	5.781	2.748	1.099	0.356	0.077
			Min	7.787	7.722	5.588	2.386	0.712	0.168	0.042
$f 6$	单模	熵	Mean	7.598	7.591	7.509	7.128	6.530	5.754	5.049
			Std	0.004	0.001	0.008	0.038	0.075	0.105	0.096
			Max	7.600	7.592	7.515	7.157	6.619	5.887	5.175
			Min	7.590	7.589	7.502	7.093	6.416	5.587	4.906
		能量	Mean	31.759	29.563	21.928	10.136	3.547	0.968	0.230
			Std	1.164	0.093	0.526	0.856	0.621	0.216	0.045
			Max	32.314	29.652	22.353	10.833	4.333	1.266	0.285
			Min	29.717	29.478	21.455	9.333	2.667	0.636	0.162
$f 7$	单模	熵	Mean	7.309	7.295	7.179	6.822	6.137	5.361	4.657
			Std	0.006	0.003	0.012	0.029	0.118	0.119	0.127
			Max	7.312	7.298	7.187	6.850	6.292	5.516	4.822
			Min	7.299	7.291	7.168	6.791	5.976	5.211	4.479
		能量	Mean	107.060	87.859	45.883	13.991	3.197	0.614	0.146
			Std	9.016	2.787	2.725	1.483	0.710	0.117	0.029
			Max	113.224	90.532	47.785	15.502	4.226	0.773	0.182
			Min	92.088	85.323	43.586	12.346	2.205	0.464	0.102

Tab. 3 Experimental results of algorithm entropy and energy under different disturbances （Rastrigin function）

$C v$	物理量	统计指标	$σ i n t$	$σ i n t$ /2	$σ i n t$ /4	$σ i n t$ /8	$σ i n t$ /16	$σ i n t$ /32	$σ i n t$ /64
2	熵	Mean	6.929	6.922	6.846	6.436	5.674	4.925	3.854
		Std	0.004	0.001	0.009	0.056	0.119	0.152	0.093
		Max	6.931	6.923	6.852	6.492	5.820	5.132	3.921
		Min	6.922	6.921	6.840	6.376	5.455	4.604	3.745
	能量	Mean	18.294	17.727	15.811	12.680	10.413	9.159	8.543
		Std	0.296	0.023	0.138	0.316	0.275	0.309	0.229
		Max	18.432	17.744	15.913	13.027	10.764	9.648	9.237
		Min	17.765	17.708	15.711	12.359	9.930	8.535	8.471
6	熵	Mean	6.929	6.922	6.851	6.532	5.774	5.162	4.251
		Std	0.004	0.001	0.007	0.069	0.181	0.122	0.001
		Max	6.931	6.923	6.857	6.610	6.023	5.291	4.251
		Min	6.922	6.920	6.846	6.437	5.476	4.848	4.250
	能量	Mean	18.215	17.697	15.993	13.346	10.660	9.183	6.433
		Std	0.287	0.027	0.123	0.414	0.417	0.182	0.003
		Max	18.355	17.719	16.101	13.897	11.260	9.447	6.435
		Min	17.731	17.675	15.892	12.780	9.982	8.774	6.433
10	熵	Mean	6.928	6.921	6.855	6.608	5.968	5.207	4.255
		Std	0.004	0.002	0.008	0.058	0.111	0.105	0.000
		Max	6.931	6.922	6.862	6.674	6.116	5.310	4.255
		Min	6.921	6.919	6.848	6.516	5.757	4.965	4.255
	能量	Mean	18.155	17.645	16.123	13.829	11.146	8.677	4.024
		Std	0.261	0.031	0.127	0.487	0.307	0.129	0.007
		Max	18.277	17.671	16.232	14.472	11.601	8.896	4.028
		Min	17.687	17.622	16.018	13.052	10.626	8.475	4.023

Tab. 3 Experimental results of algorithm entropy and energy under different disturbances （Rastrigin function）

$C v$	物理量	统计指标	$σ i n t$	$σ i n t$ /2	$σ i n t$ /4	$σ i n t$ /8	$σ i n t$ /16	$σ i n t$ /32	$σ i n t$ /64
2	熵	Mean	6.929	6.922	6.846	6.436	5.674	4.925	3.854
		Std	0.004	0.001	0.009	0.056	0.119	0.152	0.093
		Max	6.931	6.923	6.852	6.492	5.820	5.132	3.921
		Min	6.922	6.921	6.840	6.376	5.455	4.604	3.745
	能量	Mean	18.294	17.727	15.811	12.680	10.413	9.159	8.543
		Std	0.296	0.023	0.138	0.316	0.275	0.309	0.229
		Max	18.432	17.744	15.913	13.027	10.764	9.648	9.237
		Min	17.765	17.708	15.711	12.359	9.930	8.535	8.471
6	熵	Mean	6.929	6.922	6.851	6.532	5.774	5.162	4.251
		Std	0.004	0.001	0.007	0.069	0.181	0.122	0.001
		Max	6.931	6.923	6.857	6.610	6.023	5.291	4.251
		Min	6.922	6.920	6.846	6.437	5.476	4.848	4.250
	能量	Mean	18.215	17.697	15.993	13.346	10.660	9.183	6.433
		Std	0.287	0.027	0.123	0.414	0.417	0.182	0.003
		Max	18.355	17.719	16.101	13.897	11.260	9.447	6.435
		Min	17.731	17.675	15.892	12.780	9.982	8.774	6.433
10	熵	Mean	6.928	6.921	6.855	6.608	5.968	5.207	4.255
		Std	0.004	0.002	0.008	0.058	0.111	0.105	0.000
		Max	6.931	6.922	6.862	6.674	6.116	5.310	4.255
		Min	6.921	6.919	6.848	6.516	5.757	4.965	4.255
	能量	Mean	18.155	17.645	16.123	13.829	11.146	8.677	4.024
		Std	0.261	0.031	0.127	0.487	0.307	0.129	0.007
		Max	18.277	17.671	16.232	14.472	11.601	8.896	4.028
		Min	17.687	17.622	16.018	13.052	10.626	8.475	4.023

Fig. 5 Experimental results of algorithm entropy and energy for Rastrigin function

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Optimization algorithm entropy based on quantum dynamics

基于量子动力学的优化算法熵

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