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

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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 τ\tau , tt , and tˉ\bar{t} . τ\tau is the basic index, as defined in Algorithm 1 . tt denotes the time index for foptQf_{\text{opt}}^{Q} , which is updated when fQ≤foptQf^{Q}\leq f_{\text{opt}}^{Q} . tˉ\bar{t} is updated whenever fQ<foptQf^{Q}<f_{\text{opt}}^{Q} . The red line indicates the trend of fτQf_{\tau}^{Q} .
 Cost of Solution PathSample Standard DeviationImprovement Ratio 
CitiesQTZ*SAQIAQTZ*SAQIAQTZ to SAQTZ to QIAQTZ to NNNN
1001691.761732.161721.0724.3438.9342.772.333.1621.652159.27
1251920.422013.312054.5937.7157.6037.964.616.5316.432297.87
1502013.612218.522208.5049.4456.9348.419.248.8219.382497.65
1752176.762532.262617.0930.8058.1961.219.5416.838.562380.53
2002366.722924.872988.7828.2490.3662.6218.7220.4514.162769.73
Table 1: Experimental Results for TSP with More Than 100 Cities;
 IterationsImprovement RatioSolution vs Exact Minimum Ration
FunctionQTZSAQIASA/QTZQIA/QTZQTZSAQIA
Xin-She Yang N43144642017*2.04 54.57%54.57%35.22%
Salomon1727131270920.764.11100.00%99.99%99.99%
Drop-Wave25490733113.5713.04100.00%99.99%99.99%
Shaffer N22073760996573.674.66100.00%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 SetFashionMNISTCIFAR10CIFAR100STL10
ModelCNN 3 LayersResNet-50 (56 Layer Blocks)
QSLGD89.2973.837.7750.68
SGD91.4763.3125.9046.92
ASGD91.4263.4626.4347.90
QSLD91.5985.0949.6058.04
ADAM87.1282.0846.3257.32
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

Figure 3: Illustrative diagram of virtual functions used for analysis. Left: Linear virtual function — at xt\bm{x}_{t} , the gradient of the real objective function matches that of the virtual function. Right: Sinusoidal virtual function — at xt\bm{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 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 NameOSGPUCPU
PC-1Linux Ubuntu 24.0NVIDIA GeForce GTX 1080TiIntel i9 7900
PC-2Windows 11NVIDIA GeForce RTX 3050Intel i9 7900
PC-3Windows 11NVIDIA GeForce GTXTiIntel i7 6700
Appendix
Table 4: Experimental Environment
Figure 5: Comparison of 100-city TSP routes provided by each optimization algorithm
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.
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.
Figure 8: Enlarged cost trajectories from 15000 to 30000 iterations.
CriterionNarrow band α=10.0\alpha=10.0Wide band α=3.0\alpha=3.0
 SAQIAQTZSAQIAQTZ
Average minimum cost0.00310.00310.00360.0340.2160.034
Improvement ratio to the initial setting99.9399.9399.9297.7585.7397.75
Appendix
Table 5: Simulation results of the parabolic washboard potential function
FunctionQTZSAQIA
Xin-She Yang N43144642017*
Salomon172713127092
Drop-Wave2549073311
Shaffer N2207376099657
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 FunctionEquationDifficulty
Ackleyf(x)=−a⋅exp⁡(−b1d∑i=1dxi2)−exp⁡(1d∑i=1dcos⁡(c⋅xi))+a+exp⁡(1)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
Whitleyf(x)=1+14000∑i=1nxi2−∏i=1ncos⁡(xii)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 2Df(x)=∑i=1d−1[b(xi+1−xi2)2+(a−xi)2],x∈R2f(\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 100Df(x)=∑i=1d−1[b(xi+1−xi2)2+(a−xi)2],x∈R100f(\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
EggHolderf(x,y)=977−(y+47)sin⁡(∣y+0.5y+47∣)−xsin⁡(∣x−(y+47)∣)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.4f(x)=2.0+(∑i=1dsin⁡2(xi)−exp⁡(−∑i=1dxi2))exp⁡(−∑i=1dsin⁡2∣xi∣)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 FunctionAlgorithmConventional AlgorithmQTZ AlgorithmImp. ratio of StepsImp. ratio of Succ.
Final stepSuccessFinal stepSuccess  
AckleyGradient Descent843.0349.062.5013.95
Conjugate Gradient1635.0432.075.00−- 8.57
Quasi Newton(BFGS)2334.0623.073.91−- 32.35
WhitleyGradient Descent1354.01254.07.690.00
Conjugate Gradient953.0753.022.220.00
Appendix
Table 8: Experimental results of benchmark functions with 3 gradient-based search algorithms using line search
Benchmark FunctionInput DomainOptimal Point
Ackley[−32,32][-32,32][0,0][0,0]
Whitley[−512,512][-512,512][0,0][0,0]
Rosenbrock 2D[−5,10][-5,10][1,1][1,1]
Rosenbrock 100D[−5,10][-5,10][1,1,⋯ ,1]∈R100[1,1,\cdots,1]\in\mathbf{R}^{100}
EggHolder[400,600],[300,500][400,600],[300,500][522.16,413.31][522.16,413.31]
Xin-She Yang N.4[−5,5][-5,5][0,⋯0]∈R4[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.
Figure 10: Error trends of test algorithms to the dataset and neural models
Figure 11: Error trends in the performance of Adam-based QSLD with varying application periods of the enforcement function
OptimizerLearning EquationAuxiliary Equation
QSLDXτ+1Q=XτQ+Qp−1(τ)⌊Qp(τ)⋅(λh(XτQ)+r(τ,XτQ))+12⌋\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\rfloorQp=ηbpˉ(te)Q_{p}=\eta b^{\bar{p}(t_{e})}
QSLGDXτ+1Q=XτQ−Qp−1(τ)⌊Qp(τ)⋅(λ∇xf(XτQ)+r(τ,XτQ))+12⌋\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\rfloorr(τ,Xτ)=⌊λ⋅exp⁡(−ϰ(τ−τ0))1+exp⁡(−ϰ(τ−τ0))⋅h(Xτ)∥h(Xτ)∥⌋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
SGDXτ+1=Xτ−λ∇f(Xτ)\bm{X}_{\tau+1}=\bm{X}_{\tau}-\lambda\nabla f(\bm{X}_{\tau})λ=0.01\lambda=0.01
ASGDXτ+1=1t∑i=0t−1∇f(Xτ−i)\bm{X}_{\tau+1}=\frac{1}{t}\sum_{i=0}^{t-1}\nabla f(\bm{X}_{\tau-i})λ=0.01\lambda=0.01
ADAMXτ=Xτ−1−ηv^τ+ϵ⋅m^τ\bm{X}_{\tau}=\bm{X}_{{\tau}-1}-\frac{\eta}{\sqrt{\hat{\bm{v}}_{\tau}}+\epsilon}\cdot\bm{\hat{m}}_{\tau}mτ=β1⋅mτ−1+(1−β1)⋅gτvτ=β2⋅vτ−1+(1−β2)⋅gτ2m^τ=mτ1−β1τ,v^τ=vτ1−β2τ\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}
NADAMXτ=Xτ−1−ηv^τ+ϵ⋅m^τ\bm{X}_{\tau}=\bm{X}_{{\tau}-1}-\frac{\eta}{\sqrt{\hat{\bm{v}}_{\tau}}+\epsilon}\cdot\bm{\hat{m}}_{\tau}μτ=β1(1−120.96τψt))μτ+1=β1(1−120.96(τ+1)ψt)mτ=β1⋅mτ−1+(1−β1)⋅gτ,vτ=β2⋅vτ−1+(1−β2)⋅gτ2,m^τ=μτ+11−∏i=1t+1mτ+1−μτ1−∏i=1tgτ,v^τ=vτ1−β2τ,\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 AlgorithmHyperparameters
QSLDη∈219≈0.5×106,  pˉ(te)∣te=τ/B≜⌊log⁡blog⁡(τ+2)⌋\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
QSLGDEquivalent to QSLD
SGDλ=0.01\lambda=0.01
ASGDNot Defined
ADAMβ1=0.9,  β2=0.999,  ϵ=10−8.\beta_{1}=0.9,\;\beta_{2}=0.999,\;\epsilon=10^{-8}.
NADAMEquivalent to ADAM
Appendix
Table 11: Hyperparameters for Each Optimizer
Data SetFashionMNISTCIFAR10CIFAR100STL10
ModelCNN 3 LayersResNet-50 (56 Layer Blocks)
QSLGD89.2973.837.7750.68
SGD91.4763.3125.9046.92
ASGD91.4263.4626.4347.90
QSLD91.5985.0949.6058.04
ADAM87.1282.0846.3257.32
Appendix
Table 12: Experimental Results for Image Datasets: Test Accuracy for QSLGD (SGD-based) and QSLD (Adam-based).
Data SetFashionMNISTCIFAR10CIFAR100STL10
ModelCNN 3 LayersResNet-50 (56 Layer Blocks)
QSLGD0.0854260.0092530.0301040.007205
SGD0.1327470.0010420.0054782.214468
ASGD0.1309920.0011660.0049812.001648
QSLD0.0599520.0114560.0378550.005939
ADAM0.1763790.0124210.0387410.53936
Appendix
Table 13: Experimental Results for Image Datasets: Training Errors for QSLGD (SGD-based) and QSLD (Adam-based).
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