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

Jinwuk Seok; Changsik Cho

quant-phMar 12, 2026

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

Authors: Jinwuk Seok, Changsik Cho

Organizations: On-Device AIM Research Section AIC Research Lab, ETRI Daejeon, Rep. Korea

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.

Figures & tables

Figure 1: Time indices $\tau$ , $t$ , and $\bar{t}$ . $\tau$ is the basic index, as defined in Algorithm 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 $f_{\tau}^{Q}$ .

Cities	QTZ*	SA	QIA	QTZ*	SA	QIA	QTZ to SA	QTZ to QIA	QTZ to NN	NN
	Cost of Solution Path			Sample Standard Deviation			Improvement Ratio
100	1691.76	1732.16	1721.07	24.34	38.93	42.77	2.33	3.16	21.65	2159.27
125	1920.42	2013.31	2054.59	37.71	57.60	37.96	4.61	6.53	16.43	2297.87
150	2013.61	2218.52	2208.50	49.44	56.93	48.41	9.24	8.82	19.38	2497.65
175	2176.76	2532.26	2617.09	30.80	58.19	61.21	9.54	16.83	8.56	2380.53
200	2366.72	2924.87	2988.78	28.24	90.36	62.62	18.72	20.45	14.16	2769.73

Table 1: Experimental Results for TSP with More Than 100 Cities;

Function	QTZ	SA	QIA	SA/QTZ	QIA/QTZ	QTZ	SA	QIA
	Iterations			Improvement Ratio		Solution vs Exact Minimum Ration
Xin-She Yang N4	3144	6420	17*	2.04		54.57%	54.57%	35.22%
Salomon	1727	1312	7092	0.76	4.11	100.00%	99.99%	99.99%
Drop-Wave	254	907	3311	3.57	13.04	100.00%	99.99%	99.99%
Shaffer N2	2073	7609	9657	3.67	4.66	100.00%	99.99%	99.99%

Table 2: Experimental results for low-dimensional benchmark functions. Iterations denote the number of iterations each algorithm requires to find the minimum of the benchmark functions. Improvement Ratio represents the ratio of iterations required between algorithms, such as SA/QTZ and QIA/QTZ. For the benchmark function “Xin-She Yang N4”, an asterisk (*) indicates that QIA failed to find the minimum or the solution found by QTZ (100% = exact optimum).

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.

Data Set	FashionMNIST	CIFAR10	CIFAR100	STL10
Model	CNN 3 Layers	ResNet-50 (56 Layer Blocks)
QSLGD	89.29	73.8	37.77	50.68
SGD	91.47	63.31	25.90	46.92
ASGD	91.42	63.46	26.43	47.90
QSLD	91.59	85.09	49.60	58.04
ADAM	87.12	82.08	46.32	57.32

Table 3: Experimental Results for Image Datasets. Test accuracy (left) and training errors (right).

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 3 — Figure 3: Illustrative diagram of virtual functions used for analysis. Left: Linear virtual function — at $\bm{x}_{t}$ , the gradient of the real objective function matches that of the virtual function. Right: Sinusoidal virtual function — at $\bm{x}_{t}$ , the gradient of the real objective function again matches that of the virtual function.

Panel of Figure 4 — Figure 4: Conceptual diagram of the tunneling effect. Left: in the equality case, Algorithm 1 tunnels the state vector through the energy barrier at the same quantized objective value. Right: in the inequality case, Algorithm 1 tunnels the state vector through the barrier at a lower quantized objective value.

PC Name	OS	GPU	CPU
PC-1	Linux Ubuntu 24.0	NVIDIA GeForce GTX 1080Ti	Intel i9 7900
PC-2	Windows 11	NVIDIA GeForce RTX 3050	Intel i9 7900
PC-3	Windows 11	NVIDIA GeForce GTXTi	Intel i7 6700

Appendix

Table 4: Experimental Environment

Panel of Figure 5 — Figure 5: Comparison of 100-city TSP routes provided by each optimization algorithm

Panel of Figure 6 — Figure 6: Convergence trends for each algorithm in the TSP experiment. The quantization-based optimization scheme exhibits a faster convergence property compared to other algorithms. As the number of cities increases, QTZ represents better convergence performance compared to SA and QIA.

Panel of Figure 7 — Figure 7: Minimum cost trajectories for each algorithm as a function of iteration in the TSP optimization test. The minimum cost for QTZ appears much lower than that of the other algorithms because the overall cost scale for QTZ is significantly lower.

Panel of Figure 8 — Figure 8: Enlarged cost trajectories from 15000 to 30000 iterations.

Criterion	Narrow band $\alpha=10.0$			Wide band $\alpha=3.0$
	SA	QIA	QTZ	SA	QIA	QTZ
Average minimum cost	0.0031	0.0031	0.0036	0.034	0.216	0.034
Improvement ratio to the initial setting	99.93	99.93	99.92	97.75	85.73	97.75

Appendix

Table 5: Simulation results of the parabolic washboard potential function

Function	QTZ	SA	QIA
Xin-She Yang N4	3144	6420	17*
Salomon	1727	1312	7092
Drop-Wave	254	907	3311
Shaffer N2	2073	7609	9657

Appendix

Table 6: Experimental results for low-dimensional benchmark functions. Left: Number of iterations to convergence. Right: Final result as a percentage of the global optimum (100 = exact optimum, Unit is %).

Benchmark Function	Equation	Difficulty
Ackley	$f(\mathbf{x})=-a\cdot\exp(-b\sqrt{\frac{1}{d}\sum_{i=1}^{d}x_{i}^{2}})-\exp(\frac{1}{d}\sum_{i=1}^{d}\cos(c\cdot x_{i}))+a+\exp(1)$	48.25
Whitley	$f(\mathbf{x})=1+\frac{1}{4000}\sum_{i=1}^{n}x^{2}_{i}-\prod_{i=1}^{n}\cos(\frac{x_{i}}{\sqrt{i}})$	4.92
Rosenbrock 2D	$f(\mathbf{x})=\sum_{i=1}^{d-1}[b(x_{i+1}-x_{i}^{2})^{2}+(a-x_{i})^{2}],\hskip 8.19447pt\mathbf{x}\in\mathbf{R}^{2}$	44.17
Rosenbrock 100D	$f(\mathbf{x})=\sum_{i=1}^{d-1}[b(x_{i+1}-x_{i}^{2})^{2}+(a-x_{i})^{2}],\hskip 8.19447pt\mathbf{x}\in\mathbf{R}^{100}$	None
EggHolder	$f(x,y)=977-(y+47)\sin\left(\sqrt{\|y+0.5y+47\|}\right)-x\sin\left(\sqrt{\|x-(y+47)\|}\right)$	18.92
Xin-She Yang N.4	$f(\mathbf{x})=2.0+\left(\sum_{i=1}^{d}\sin^{2}(x_{i})-\exp(-\sum_{i=1}^{d}x_{i}^{2})\right)\exp(-\sum_{i=1}^{d}{\sin^{2}\sqrt{\|x_{i}\|}})$	26.33

Appendix

Table 7: Benchmark functions and their optimization difficulty scores for locating global optima.

Benchmark Function	Algorithm	Conventional Algorithm		QTZ Algorithm		Imp. ratio of Steps	Imp. ratio of Succ.
Benchmark Function	Algorithm	Final step	Success	Final step	Success
Ackley	Gradient Descent	8	43.0	3	49.0	62.50	13.95
	Conjugate Gradient	16	35.0	4	32.0	75.00	$-$ 8.57
	Quasi Newton(BFGS)	23	34.0	6	23.0	73.91	$-$ 32.35
Whitley	Gradient Descent	13	54.0	12	54.0	7.69	0.00
	Conjugate Gradient	9	53.0	7	53.0	22.22	0.00

Appendix

Table 8: Experimental results of benchmark functions with 3 gradient-based search algorithms using line search

Benchmark Function	Input Domain	Optimal Point
Ackley	$[-32,32]$	$[0,0]$
Whitley	$[-512,512]$	$[0,0]$
Rosenbrock 2D	$[-5,10]$	$[1,1]$
Rosenbrock 100D	$[-5,10]$	$[1,1,\cdots,1]\in\mathbf{R}^{100}$
EggHolder	$[400,600],[300,500]$	$[522.16,413.31]$
Xin-She Yang N.4	$[-5,5]$	$[0,\cdots 0]\in\mathbf{R}^{4}$

Appendix

Table 9: Input domain (represented with min and max per component) and optimal point corresponding to benchmark functions

Figure 18

Figure 9: Results of successful search traces for each algorithm on benchmark functions. For a fair comparison, we selected search results in which both the conventional and quantization-based algorithms successfully found an optimal point. The blue line indicates the search trajectories of conventional algorithms, while the green line shows those of the quantization-based algorithm. In some cases, the conventional search trajectory appears as a straight line, which may look like the result of a single iteration; however, these actually required multiple iterations along the straight path. In contrast, the quantization-based algorithm, which produces bent trajectories, required fewer iterations compared to the conventional algorithms.

Panel of Figure 10 — Figure 10: Error trends of test algorithms to the dataset and neural models

Panel of Figure 11 — Figure 11: Error trends in the performance of Adam-based QSLD with varying application periods of the enforcement function

Optimizer	Learning Equation	Auxiliary Equation
QSLD	$\bm{X}_{\tau+1}^{Q}=\bm{X}_{\tau}^{Q}+Q_{p}^{-1}(\tau)\left\lfloor Q_{p}(\tau)\cdot\left(\lambda h(\bm{X}_{\tau}^{Q})+r(\tau,\bm{X}_{\tau}^{Q})\right)+\frac{1}{2}\right\rfloor$	$Q_{p}=\eta b^{\bar{p}(t_{e})}$
QSLGD	$\bm{X}_{\tau+1}^{Q}=\bm{X}_{\tau}^{Q}-Q_{p}^{-1}(\tau)\left\lfloor Q_{p}(\tau)\cdot\left(\lambda\nabla_{\bm{x}}f(\bm{X}_{\tau}^{Q})+r(\tau,\bm{X}_{\tau}^{Q})\right)+\frac{1}{2}\right\rfloor$	$r(\tau,\bm{X}_{\tau})=\left\lfloor\lambda\cdot\frac{\exp(-\varkappa(\tau-\tau_{0}))}{1+\exp(-\varkappa(\tau-\tau_{0}))}\cdot\frac{h(\bm{X}_{\tau})}{\\|h(\bm{X}_{\tau})\\|}\right\rfloor$
SGD	$\bm{X}_{\tau+1}=\bm{X}_{\tau}-\lambda\nabla f(\bm{X}_{\tau})$	$\lambda=0.01$
ASGD	$\bm{X}_{\tau+1}=\frac{1}{t}\sum_{i=0}^{t-1}\nabla f(\bm{X}_{\tau-i})$	$\lambda=0.01$
ADAM	$\bm{X}_{\tau}=\bm{X}_{{\tau}-1}-\frac{\eta}{\sqrt{\hat{\bm{v}}_{\tau}}+\epsilon}\cdot\bm{\hat{m}}_{\tau}$	$\begin{aligned} \bm{m}_{\tau}&=\beta_{1}\cdot\bm{m}_{\tau-1}+(1-\beta_{1})\cdot\bm{g}_{\tau}\\ \bm{v}_{\tau}&=\beta_{2}\cdot\bm{v}_{\tau-1}+(1-\beta_{2})\cdot\bm{g}_{\tau}^{2}\\ \hat{\bm{m}}_{\tau}&=\frac{\bm{m}_{\tau}}{1-\beta_{1}^{\tau}},\\ \hat{\bm{v}}_{\tau}&=\frac{\bm{v}_{\tau}}{1-\beta_{2}^{\tau}}\end{aligned}$
NADAM	$\bm{X}_{\tau}=\bm{X}_{{\tau}-1}-\frac{\eta}{\sqrt{\hat{\bm{v}}_{\tau}}+\epsilon}\cdot\bm{\hat{m}}_{\tau}$	$\begin{aligned} \mu_{\tau}&=\beta_{1}\left(1-\frac{1}{2}0.96^{\tau\psi_{t}})\right)\\ \mu_{\tau+1}&=\beta_{1}\left(1-\frac{1}{2}0.96^{(\tau+1)\psi_{t}}\right)\\ \bm{m}_{\tau}&=\beta_{1}\cdot\bm{m}_{\tau-1}+(1-\beta_{1})\cdot\bm{g}_{\tau},\\ \bm{v}_{\tau}&=\beta_{2}\cdot\bm{v}_{\tau-1}+(1-\beta_{2})\cdot\bm{g}_{\tau}^{2},\\ \hat{\bm{m}}_{\tau}&=\frac{\mu_{\tau+1}}{1-\prod_{i=1}^{t+1}}\bm{m}_{\tau}+\frac{1-\mu_{\tau}}{1-\prod_{i=1}^{t}}\bm{g}_{\tau},\\ \hat{\bm{v}}_{\tau}&=\frac{\bm{v}_{\tau}}{1-\beta_{2}^{\tau}},\\ \end{aligned}$

Appendix

Table 10: Learning Equations for Each Optimizer

Learning Algorithm	Hyperparameters
QSLD	$\eta\in 2^{19}\approx 0.5\times 10^{6},\;\bar{p}(t_{e})\|_{t_{e}=\tau/B}\triangleq\lfloor\log_{b}\log(\tau+2)\rfloor$
QSLGD	Equivalent to QSLD
SGD	$\lambda=0.01$
ASGD	Not Defined
ADAM	$\beta_{1}=0.9,\;\beta_{2}=0.999,\;\epsilon=10^{-8}.$
NADAM	Equivalent to ADAM

Appendix

Table 11: Hyperparameters for Each Optimizer

Data Set	FashionMNIST	CIFAR10	CIFAR100	STL10
Model	CNN 3 Layers	ResNet-50 (56 Layer Blocks)
QSLGD	89.29	73.8	37.77	50.68
SGD	91.47	63.31	25.90	46.92
ASGD	91.42	63.46	26.43	47.90
QSLD	91.59	85.09	49.60	58.04
ADAM	87.12	82.08	46.32	57.32

Appendix

Table 12: Experimental Results for Image Datasets: Test Accuracy for QSLGD (SGD-based) and QSLD (Adam-based).

Data Set	FashionMNIST	CIFAR10	CIFAR100	STL10
Model	CNN 3 Layers	ResNet-50 (56 Layer Blocks)
QSLGD	0.085426	0.009253	0.030104	0.007205
SGD	0.132747	0.001042	0.005478	2.214468
ASGD	0.130992	0.001166	0.004981	2.001648
QSLD	0.059952	0.011456	0.037855	0.005939
ADAM	0.176379	0.012421	0.038741	0.53936

Appendix

Table 13: Experimental Results for Image Datasets: Training Errors for QSLGD (SGD-based) and QSLD (Adam-based).

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Abstract

HJB Equations

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