Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 12, No. 3, 2025 

 

36 

Research	on	Machine	Learning	Optimization	Algorithm	
Based	on	QUBO	Model	
Minyi Ye *, Lanqing Wu, Guanquan Zhu, Jiaguo Jiang  

School of Mathematics and Information Science, Guangzhou University, Guangzhou Guangdong, 510006, China 

* Corresponding author: Minyi Ye (Email: 32215150090@e.gzhu.edu.cn) 

 

Abstract: This paper deals with the research of QUBO model optimization machine learning algorithm, including the 
following: first, discretize the continuous parameters of the machine learning model into binary variables, and construct the 
QUBO objective function; then optimize the QUBO model by using the quantum annealing algorithm; and finally solve the 
globally optimal solution by adjusting the annealing parameters, which include the initial temperature, the coefficient of 
temperature reduction, and the number of iterations. QUBO model can discretize continuous variables (including parameters 
such as weights and bias) in AR, SVM, CNN and other models in machine learning into binary variables, so as to better deal 
with nonlinear relationships and reduce the computational complexity in the training process. With the QUBO model, the 
regularization term can be better controlled to avoid the overfitting problem. Meanwhile, the QUBO model can be solved in 
parallel by quantum computing or simulated annealing algorithm, which significantly improves the computational efficiency 
under large-scale data. 

Keywords: QUBO; AR; SVM; CNN; Quantum Computing; Simulated Annealing Algorithm. 
 

1. Introduction 
The development of quantum technology is of great 

scientific significance and social value, which is expected to 
have a major impact on traditional technology and trigger 
technological revolution and industrial transformation [1]. 

Among them, quantum computing stands out in decision 
problems, classification problems, as well as prediction 
problems. In recent years, people have discovered that a 
mathematical formulation known as QUBO, an acronym for 
a Quadratic Unconstrained Binary Optimization problem. 
Through special reformulation techniques that are easy to 
apply, the power of QUBO solvers can be used to efficiently 
solve many important problems once they are put into the 
QUBO framework [2]. 

According to the above description and relevant data, this 
paper built mathematical models to solve the following 
problems. 

(1) Transform the Autoregressive (AR) model used in the 
time series forecasting problem whose primitive data ranging 
from January to September into a QUBO model and define 
the objective function and decision variables clearly. Based 
on this, use Kaiwu SDK's simulated annealing algorithm to 
solve the model and predict the demand for October. 

(2) Classify the provided dataset using an SVM model, 
among which you should transform the optimization problem 
of training an SVM-based classification model into a QUBO 
model and define the objective function and decision 
variables. Besides, solve the QUBO problem using the 
simulated annealing algorithm in the Kaiwu SDK. 

(3) Explore the integration of quantum computing and deep 
learning by selecting a specific application scenario like 
image classification or recommendation systems and design a 
deep learning model such as Convolutional Neural Net-works 
(CNNs), then convert the model's training optimization into a 
QUBO form, Finally use the simulated annealing algorithm 
to solve this optimization problem. 

2. Model 1: The QUBO Model Based 
on Time Series Prediction 

(1) Establishment of QUBO Resource Demand Prediction 
Model Based on AR 

Firstly, based on the given problem settings, we conduct 
predictions by employing the autoregressive model. 
Autoregressive (AR) model is a model for studying time 
series, with wide applicability and high accuracy. 

AR model is mainly based on the linear combination of past 
observations and present disturbance values for prediction. 
The general P-order autoregressive process AR (p) is[3] 

∑              (1) 
Where, Y indicates the predicted value of resource demand 

in month t, and Y  indicates the real value of resource 
demand in month (t-1); c  is a constant, representing the 
average level of the time series; φ   refers to the 
autoregressive model parameter, representing the effect of the 
i-th lag term on the current value; ε  refers to random noise 
with a mean of zero; p is the order of the AR model, indicating 
how many lag terms are used to predict the current value, as 
determined by the Akaike information criterion (AIC) and 
p=3. 

On this basis, We transform the above time series 
prediction problem into Quadratic Unconstrained Binary 
Optimization (QUBO) model. QUBO, whose minimum 
energy state is called the ground state. The QUBO model 
consists of multiple 0-1 binary variables, and a combination 
of which gives one solution. In particular, the combination of 
binary variables giving the minimum energy state of the 
QUBO model is called the ground-state solution. 

A QUBO model is represented by the formula below using 
a binary variable  that takes 0 or 1( 0	 	 1). 

∑ ∑           (2) 
Where ， ⋯  is a set of binary variables; 	is 

the external magnetic field coefficient representing the force 
on the binary variable  ; and   is the interaction 



 

37 

coefficient representing the weight of the connection between 
the binary variables  and . A solution of a QUBO model 
is one of the combinations of binary variables. The ground-
state solution of the QUBO model is the combination of 
binary variables that minimizes . The value of  for 
a given combination of binary variables is called the objective 
value. 

Remember that Q is composed of a matrix , according 
to the QUBO Hamiltonian produced by it, the objective 
function is transformed into a quadratic form, then formula (2) 
can be simplified as: 

               (3) 
Where x is the binary decision variable,  represents the 

transpose of ,	a is the linear term coefficient vector, Q is the 
quadratic term coefficient matrix of the quadratic 
unconstrained binary optimization problem, and it is a 
symmetric matrix, otherwise the form of the coefficient 

matrix Q can be changed by the following formula: 
Symmetrical	Form								

Upper	Triangular	Form															
	 , 	

	 0,

 (4) 

We take 70% of the data as the training set, 30% of the data 
as the test set, by determining the Sum of Squared Errors (SSE) 
of the predicted value of the minimum autoregressive 
function, predict the Computing resource demand in October. 
In this problem, we aim to minimize the predicted value error 
while satisfying the constraint that demand varies within a 
reasonable range. Therefore, based on the above Analysis, an 
objective function D with minimized predicted value is 
constructed: 

∑            (5) 
Combined with formula (3), we can see that the objective 

function can be expressed as follows: 

 

                (6) 

 

Based on this, we define a (1+p) · n-dimensional vector x 
to discretize the continuous values c and   to binary 
decision variables, where :  is the binary representation of 
c and the binary representation of   is :  , 

arranged in the order of i: 
, , ⋯ , , ,⋯ , ,⋯     (7) 

The mathematical formula for the specific binarization is 
as follows: 

 
∑ 2

∑ 2
                                  (8) 

 
Substituting formula (8) into formula (6) can simplify and expand the objective function: 

 
∑   2 ∑   2 2 ∑   ∑   2

∑   2 2 ∑   2 ∑   ∑   2

∑   ∑   2 	

               (9) 

 

And since the constant term has nothing to do with the 
value of the decision variable when the function D gets the 
minimum value, for simple operation, we directly ignore the 

constant term  , formula (9) can be simplified to the 
following mathematical expression: 

 
∑   2 ∑   2 2 ∑   ∑   2 		

∑   2 2 ∑   2 ∑   ∑   2

∑   ∑   2

                      (10) 

 
In addition, we constrain the decision variable by the following formula: 

 
0 2 1
2 1

                                        (11) 

 
Among them, by observing the demand data from January 

to September, we calculate and find that the data 9000 in 
January is close to 2 1, so the value range of c is set in 
the interval 0, 2 1  ; Similarly, we find that the 
difference of data multiples is not more than two times, so the 
value range of  is set in the interval 2 , 1 . 

Among them, the QUBO matrix needs to be converted to 

Ising variable before using Kaiwu SDK. The Ising Model is a 
random process model that describes the phase transition of 
matter, which is abstracted into the following mathematical 
form: 

∑ , ∑         (12) 
Where,  is the spin variable to be found, the value is {-

1,1}, H is the Hamiltonian, J is the quadratic term coefficient, 



 

38 

 and h is the linear term coefficient, are known quantities[4] 
We adopt the following formula as the conversion formula 

between the binary variable and the Ising variable: 
	                   (13) 

Where, 1	 correspond 0 , 1 
correspond	 1. 

Here, we summarize the solution of this model: 

 
Figure 1. The solution of model 1 

 
(2) Solution of Resource Demand Prediction Model Based 

on SA 
In order to solve the resource demand corresponding to 

October when the error of the predicted value is minimized, 
we adopted the simulated annealing solver built in Kaiwu 
SDK to solve it. The following is the flow chart of the 
simulated annealing algorithm solver. 

 

 
Figure 2. Flow chart of QUBO model solved by simulated 

annealing 
 
For the parameters in the simulated annealing solver in the 

Kaiwu SDK, we set them as follows: 
Select the initial temperature  and the end temperature 

, and select the appropriate cooling coefficient  during 
the annealing process, so that the iterative depth of each 
temperature in the annealing process does not exceed 100. 

 
Table 1. Simulated annealing parameter table 
Parameters Select parameter values

Initial temperature  3000 
Cutoff temperature  10  

Cooling factor  0.90 
Iterations per temperature 100 

Size limit 100 
 

By solving the simulated annealing solver, we get the 
following results: 

 
Table 2. Optimal prediction parameter values table 

Optimal prediction 
parameters

Value 

c 1393 

 0.6484 

 0.1016 

 0.1351 
 

At this point, the mean square error   has reached its 
minimum value of 30,723.78. Among them,   can be 
regarded as the weight of the real value i before the predicted 
value to the predicted value, then the phenomenon of 
	great influence on the predicted value is in line with reality, 

which is reasonable. 
On this basis, we substitute the formula (1) to solve the 

predicted value of each month as follows: 
 

Table 3. Statistics of predicted values for each month 

Month 
Predicted 

Value
 Month 

Predicted 
Value

April 9784.98  August 10443.38 

May 10030.58  September 10613.51 

June 10147.90  October 10719.36 

July 10248.88    
 

In addition, we draw the fitting image as follows: 
It can be seen from the above figure that the predicted 

results have a high goodness of fit. Based on this, we 
calculated a variety of prediction evaluation indicators to help 
explain the excellent performance of the results. 

In addition to this, we also visualized the process of 
Hamiltonian change over time in the simulated annealing 
algorithm to reflect the energy state of the system. 

As can be seen from the figure above, when the system is 
in the initial state, the energy is higher; Then, the Hamiltonian 
decreases rapidly, and the system quickly finds a lower energy 
state in the optimization process; When the Hamiltonian 
approaches 0.03 seconds, there is a small increase, which is 
because the algorithm is trying to jump out of the local 
optimal solution. After the decrease, the Hamiltonian tends to 
be stable, indicating that the algorithm has converged and the 
system has reached the global optimal state. Solving time of 
the QUBO model is approximately 0.07s. 

3. Model 2: The QUBO Model Based 
on SVM Classification 

(1) Establishment of QUBO-Classification Model Based 
on SVM 



 

39 

 

 
Figure 3. The predicted value compared with the true value 

 
Table 4. Results evaluation index value statistics 

Prediction Evaluation Indicators Value 
MAE 67.624 

MAPE 0.667 
 0.933 

 
Figure 4. Visualization of Hamiltonian value over time 

 
Support Vector Machine (SVM) is developed from the 

optimal classification surface in the case of linear separability. 
The basic idea can be illustrated by the two-dimensional 
situation in Figure 4. In the figure below, the solid point and 
the hollow point represent two types of samples respectively. 
If they are linearly separable, the result of machine learning 
is a hyperplane (also known as the discriminant function), 
which divides the training sample into positive and negative 
classes. 

Obviously, there are infinite number of such hyperplanes. 
According to the requirement of the Structure Risk 
Minimization principle (SRM), the result of machine learning 
should be the optimal hyperplane, which can not only 
correctly separate the two types of training samples. but also 
maximizes the classification interval. Since SVM is a binary 

classification model, while this is a three-classification 
problem, we first split the three-classification problem into 
three different binary classification problems. including 
distinguishing between Setosa and Versicolor, distinguishing 
between Setosa and Virginica, including distinguishing 
between Setosa and versicolor, distinguishing between Setosa 
and virginica, distinguishing between Virginica and 
Versicolor. Distinguishing between Virginica and Versicolor. 
By doing this, a tripartite classification problem is 
transformed into three independent binary classification 
problems. 

Suppose an M-dimensional hyperplane is described by the 
following equation [5]: 

∙ 0			 ∈ , ∈ 		       (14) 
 



 

40 

 
Figure 5. Classification hyperplane in the linearly separable case 

 
Then the optimal hyperplane with the largest classification 

interval can be obtained by finding the minimum value of 
‖ ‖

, where the constraints are: 

∙ 1 0			 1,⋯ , 		     (15) 
 

Where, ∈ 1, 1 ,  	is the sample eigenvector. The 
constraint conditions ensure that the distance from the data 
point to the hyperplane is at least 1. When the above formula 
obtains an equal sign, it means that the data point is exactly 
on the interval boundary, and these data points are called 
support vectors. 

In the case of linear indivisibility, such as the presence of 
noisy data, SVM introduces a relaxation term 0 in the 
above equation to achieve soft spacing, i.e.: 

∙ 1 				 1,⋯ , 		   (16) 
Based on this, the objective function is changed into the 

following formula: 
min , ∑ 		    (17) 

Where w is the normal vector of the hyperplane, b is the 
bias of the hyperplane and 	is the relaxation variable, C is 
the regularization parameter, which is used to control the 
model's tradeoff between classification interval and 
misclassification, and determines the relative importance of 
the two, while its value is selected by cross-validation, that is, 
adjusting C according to the performance on the verification 
set. 

Based on the above analysis, we transform SVM into 
QUBO model, which means that formula (17) is transformed 
into formula (3). 

First, we discretized w, b, and  in SVM by converting to 
binary. Suppose 	 ∈ ,  , 	 represented by m 
binary variables , ∈ 0,1 : 

∆ ∑ , ∙ 2 	      (18) 
Of which, 

∆               (19) 

Then the mathematical expression of binary conversion of 
w binormal is as follows: 

‖ ‖ ∑ ∑ ∆ ∑ , ∙ 2  (20) 
In the same way, we assume that ∈ ,  , 

represented by  binary variable , ∈ 0,1 : 

∆ ∑ , ∙ 2 	        (21) 
Among them, 

∆ 		               (22) 

And , 	 respectively represent the minimum and 
maximum of b. 

To convert formula (15) into QUBO penalty term, we have: 

∑ max 0,1 		 (23) 
Where, ∑ , ,  b is the bias and   is the 

penalty term weight. 
Secondly, we substitute the binary expressions of w and b 

into the above formula respectively, then we have: 

     (24) 

 

Where, ∑   ,   is the constant term, 

∆ ∑   ∑   , ⋅ , ⋅ 2 ∆ ∑   , ⋅ 2   is the 

part about the binary variable. Finally, substitute the above 
into the formula (23): 

              (25) 

 
Where 1 ∑   ,  is the constant term, 

∆ ∑   ∑   , , 2 ∆ ∑   , 2   is the 
linear term about the binary variable, respectively remember 
the constant term is ,  , then the above formula can be 

abbreviated as follows: 
1 	          (26) 

Then formula (23) can be written as: 
∑ max 0, 		     (27) 



 

41 

Consequently, the objective function can be integrated as follows: 

                  (28) 

 

(2) Solving QUBO-Classification Model Based on SA 
Firstly, we do data preprocessing, which scramps the 

original ordered data, and uses 70% of the data for data 
training, and the rest of which is used for data prediction. 

We still use the simulated annealing solver built in Kaiwu 
SDK for solving, in which the parameters in the solver are set 
as follows: 

 
Table 5. Simulated annealing parameter table 

Parameters Select parameter values 

Initial temperature  100 

Cutoff temperature  10  

Cooling factor  0.99 

Iterations per temperature 10 

Size limit 10 

 
Select the initial temperature  and the end temperature 

, and select the appropriate cooling coefficient  during 
the annealing process, so that the iterative depth of each 
temperature in the annealing process does not exceed 10. 

By solving the simulated annealing solver, we get 3 
independent binary classification problem parameter results: 

 
Table 6. Optimal prediction parameter values (Iris-setosa and Iris-

versicolor) 

Optimal prediction parameters Value 

 -0.269 

 0.338 

 -0.701 

 -0.749 

b -0.247 

 50 

 
At this point, the objective function Q reaches a minimum, 

and w is the decision hyperplane normal vector, and b is the 
bias and  is the penalty term weight. 

On this basis, we draw the classification result image as 
follows: 

It can be seen from the above figure that QUBO-SVM 
binary classification results are more accurate. Based on this, 
we calculated a variety of prediction evaluation indicators to 
help explain the excellent results. 

 

 

 
Figure 6. Comparison of classical SVM and QUBO-SVM 

classification (Iris-setosa and Iris-versicolor) 
 

 
Table 7. Statistics of outcome evaluation indicators (Iris-setosa and 

Iris-virginica) 

Data set 
Recall 

rate
F1 

Accuracy 
Rate 

Precision 
rate

Training Set 1 1 1 1
Test Set 1 1 1 1
 

Based on this, we solve the hyperplane parameters of Iris-
setosa and Iris-virginica: 

 
 
 
 
 
 
 



 

42 

Table 8. Optimal prediction parameter values (Iris-setosa and Iris-virginica) 

Optimal prediction parameters Value 

 -0.098 

 0.138 

 -0.663 

 -0.657 

b -0.139 

 24 

At this point, the objective function Q reaches its minimum 
value. Based on this, we draw the classification result image 
as follows: 

 

 

 
Figure 7. Comparison of classical SVM and QUBO-SVM 

classification (Iris-setosa and Iris-virginica) 
 

As can be seen from the above figure, QUBO-SVM binary 
classification results are more accurate. Based on this, we 
calculate multiple prediction evaluation indicators to assist in 
illustrating the excellent results. 

 
Table 9. Statistics of outcome evaluation indicators (Iris-setosa and 

Iris-virginica) 

Data set 
Recall 
Rates 

F1 
Accuracy 

rate 
Precision 

rate
Training Set 1 1 1 1

Test Set 1 1 1 1
 

In the same way, we solve for the hyperplane parameters of 
Iris-versicolor and Iris-virginica: 

 
 
 
 

Table 10. Optimal prediction parameter values (Iris-versicolor and 
Iris-virginica) 

Optimal prediction parameters Value 

 1.026 

 2.191 

 -5.459 

 -7.785 

b 0.794 

 10  
 

At this point, the objective function Q reaches the 
minimum value, where, due to the fusion of data points of 
Iris-versicolor and Iris-virginica, we draw a comparison 
diagram of classification results of Iris-versicolor and Iris-
virginica between SVM and QUBO-SVM: 

 

 

 
Figure 8. Comparison of classification results between classical 

SVM and QUBO-SVM (Iris-versicolor and Iris-virginica) 
 

In addition, we calculated a variety of predictive evaluation 
indicators to assist in illustrating superior classification 



 

43 

performance:  
 

Table 11. Statistics of outcome evaluation indicators (Iris-versicolor and Iris-virginica) 
Data set Recall Rates F1 Accuracy Rate Precision rate

Training Set 0.957 0.957 0.957 0.958
Test Set 0.942 0.933 0.933 0.933

In addition, we also visualized the change of Hamiltonian 
over time in the simulated annealing algorithm during the 

solving of 3 binary classification problems to reflect the 
energy state of the system. 

 

 

 
Figure 9. Visualization of Hamiltonian changes over time for 3 binary classification problems 

 
From the figure above, we can see that the Hamiltonian 

drops rapidly at the beginning, which indicates that the energy 
of the system decreases rapidly, and then there are some 
fluctuations. These fluctuations indicate that during the 
optimization process, the system is exploring different states 
in order to find the state with lower energy. As time goes on, 
the rate of decline gradually slows down, and the Hamiltonian 
tends to stabilize, which may mean that the system has 
approached or reached a stable state. And Solving time of the 
QUBO model is between 0.07s and 0.09s. And solving time 
of the QUBO model is between 0.07s and 0.09s. 

4. Model 3: The QUBO Model Based 
on CNN Image Classification 

(1) Establish QUBO Model of CNN for Image 
Classification of MNIST Handwritten Numerals Dataset 

So far, the pattern recognition system based on 
convolutional neural network is one of the systems with the 
best performance, especially in the field of handwritten 
character recognition, and has been used as the evaluation 

standard for the performance of machine recognition systems. 
By mining the spatial correlation in the data, convolutional 
neural network can reduce the number of trainable parameters 
in the network, so as to improve the backpropagation 
algorithm efficiency of the forward propagation network. 

As shown in the following figure, after the input layer reads 
images of a unified size, the input image is convolved with 
three filters and one additive bias. After convolution, three 
feature maps are generated in layer C1. Then, the four 
adjacent pixels in the feature map are grouped together to 
obtain the average value, and then weighted value and bias 
are added. Three feature maps of layer S2 are obtained by 
ReLU activation function. 

Among them, 
max 0, 29  

These maps are then filtered accordingly to get the C3 layer. 
This layer then produces S4, as does S2. Finally, these pixel 
values are rasterized, connected into a one-dimensional vector 
input into a traditional neural network, and then enter the fully 
connected layer to produce an output. 

 
Figure 10. Illustration of convolutional neural networks 



 

44 

 
Among them, layer C is the convolutional layer, that is, the 

feature extraction layer. A convolutional layer usually 
contains multiple feature graphs with different weight vectors, 
so that a variety of different features can be obtained at the 
same position. Layer S is the pooling layer, which reduces the 
resolution of the feature map and reduces the sensitivity of the 
network output to displacement and deformation through 
local averaging and downsampling operations. Subsequent 
convolutional layers and pooling layers are alternately 
distributed and connected, forming a "double pyramid" 
structure: the number of feature maps gradually increases, and 
the resolution of the feature maps gradually decreases. 

Convolutional layers are the core of CNN, and each 
convolutional layer uses several convolutional layers to scan 
the input image. As for every convolution kernel, the 
convolution operation computes the weighted sum of the local 
region between the input image and the convolution kernel. 
Assuming the size of the input image is   (height, 
width, depth), the convolution operation uses a convolution 
kernel of size , depth D, and the number of convolution 
cores (number of output channels) is F. For each local region 

  and convolution kernel W in the input image X, the 
convolution operation is as follows [9]: 

      , , , , 	 30  

Where, the convolution kernel , , 	representing the (m, 
n, d) element, , ,  is the (i+m-1, j+n-1, d) pixel 
of the input image, and b is the offset term. 

The pooling layer is used to downsample the feature map 
output of the convolutional layer. The pooling operation 
performed here is the maximum pooling, that is, for each 

 region in the feature map, the maximum value in the 
region is selected as the output. 

Each image in the MNIST dataset corresponds to a digital 
label from 0 to 9, which is composed of NIST Special 
Database 1 and NIST Special Database 3. The training set in 
MNIST selects 30,000 samples from SD-3 and SD-1 
respectively. The 60,000 samples were drawn from 
handwritten data from approximately 250 different 
individuals, and the 5,000 samples each from SD-3 and SD-1 
were similarly selected as the test set. The images in the 
MNIST dataset are all of size, and the images of some 
numbers in the dataset are28 28[6] 

 
Figure 11. Images of the first 16 numbers in the MNIST data set 

 
In the multi-class classification problem, the objective 

function is minimize the difference between the predicted 
result and the true label. Assuming that the model predicts the 
result is  and that the actual label is , we use the cross 
entropy loss function to measure the error [7]: 

,   log 	 31  

Where C is the total number of categories, and for this 
dataset, C is the constant 10; is the k-th element of the 
sample  's true label, if the sample belongs to class k, 

1 , then 0 otherwise;  is the probability that the 
sample   predicted by the model belongs to class k. The 
output layer of CNN uses softmax activation function to 
calculate the probability of each class, softmax activation 
function is the mathematical expression as follows[8]: 

∑
32  

Where,  is the raw score of class k output by the fully 

connected layer,  	is the predicted probability of class k. 
For the convolutional layer l and the fully connected layer, 

the output can be expressed as follows: 
ReLU ∗ 33  

Where,   refers to the weight matrix of the 
convolution layer and the fully connected layer, and X is the 
output (or input image) of the previous layer and  is the 
biased term. In addition,  denoted as the target value, then 
the error term of the convolution layer can be expressed as 
follows: 

	 34  

In order to combine the variables involved in the above 
theoretical Analysis with QUBO, we defined  as the binary 
activation variable of the i-th neuron, indicating whether the 
neuron is activated, with the value of 0 or 1; Binary variable 

  for the model prediction class k, indicating whether the 
sample is classified as class k, taking the value 0 or 1; In 
addition, we need to replace it with binary ,   to 
expression. 

To prevent overfitting, we add regularization terms, 
especially L2 regularization terms. 

,

	 35  

Based on the above Analysis, we determine that the 
objective function is: 

(36) 

Where, , ,   is the penalty 
coefficient that controls the importance of the error term. 

On this basis, we determine the constraint conditions, the 
classification results should meet "only one category is 
predicted", the activation value of neurons can only be 0 or 1, 
the discretization into binary decision variables in 
convolution operation, the calculation rules of weighted sum 
of neurons, the minimization of cross entropy loss function 
and regularization constraints, the specific mathematical 
expressions are as follows: 



 

45 

  1

∈ 0,1 , ∀

conv  

fe   fe fe

∈ , , … , , ∀ ,

cross-entropy   log

regularization reg  
,

37  

Where,   is the predicted binary variable of class k, 
, , … ,  is the weight value after discretization. 

(2) Solution of Image Recognition Classification Codel 
Based on Simulated Annealing 

We still use the simulated annealing solver built in Kaiwu 
SDK for solving, in which the parameters in the solver are set 
as follows: 

 
Table 12. Simulated annealing parameter table 

Parameters Select parameter values
Initial temperature  3000 

Cutoff temperature  10  
Cooling factor  0.90 

Iterations per temperature 100 
Size limit 100 

 
Select the initial temperature  and the end temperature 

, and select the appropriate cooling coefficient  during 
the annealing process, so that the iterative depth of each 
temperature in the annealing process does not exceed 100. 

Through solving, we captured part of the images in the 
MNIST dataset and their recognition results, as shown below: 

 

 
Figure 12. Part of the image recognition results 

 
In addition to this, we visualize the changes in the loss (left) 

and accuracy (right) of the model training process over the 
training cycle: 

 

 
Figure 13. The change of loss and accuracy in the model training process with the training cycle 

 
As can be seen from the figure on the left, both training 

losses and validation losses decrease as the training cycle 
increases, indicating that the model is learning and gradually 
reducing the prediction error. After the initial decline, the 
validation loss levels off and is always lower than the training 
loss, indicating that the model performs well on the training 
set without significant overfitting. 

For the figure on the right, both the training accuracy and 
validation accuracy rise as the training cycle increases, and 
both approach 1.0, indicating that the model's predictions are 
becoming more accurate. And the training accuracy and 
verification accuracy are very close, which further indicates 

that the model is not overfitting and has good generalization 
ability on both the training set and the verification set. 

In addition, we calculated a variety of predictive evaluation 
indicators to help explain the excellent performance of image 
recognition classification results: 

 
Table 13. Results evaluation index value statistics (Iris-versicolor 

and Iris-virginica) 

Data set recall 
F1-

score 
accuracy support 

Macro avg 0.99 0.99 0.99 10000
Weighted avg 0.99 0.99 0.99 10000

 



 

46 

From the table above, we can see that the accuracy, recall 
rate, and F1 score of the model recognition results are close 
to 0.99 or 1.00, indicating that the model performs very well 
on handwritten digit recognition tasks. solving time of the 
QUBO model is 211s due to the huge amount of data and 
individual differences in hardware performance. 

5. Conclusion 
For model 1, we used the monthly computational resource 

demand data provided by the problem and combined with 
AIC criterion to establish an autoregressive model of order 3. 
To predict the data in October, we took predicted resource 
demand in October as the result when SSE minimises. Then 
AR model is transformed into QUBO, and the parameters in 
AR are replaced by binary expressions, and the SSE function 
of binary variable as the decision variable is taken as the 
objective function. The minimum value is obtained when SSE 
equals to 30723.78, and the predicted value in October is 
10719.36. In addition, we calculated that the MAPE of the 
model is 0.709, and the R-squared is 0.933, indicating that the 
prediction result of the model was excellent. By using Kaiwu 
SDK, the solving time of the QUBO model was 0.07 seconds. 

For model 2, we decomposed the tripartite classification 
into three binary classification tasks: Setosa vs. Versicolor, 
Setosa vs. Virginica, and Virginica vs. Versicolor. The 
objective function minimizes the norm of the normal vector 
of the SVM hyperplane for each task. The normal vector, bias, 
and relaxation variables of the hyperplane are discretized into 
binary variables, with a penalty term and weight introduced. 
The objective is then transformed into a QUBO, solved using 
the Kaiwu SDK, and evaluated. Here only demonstrate the 
normal vector for Setosa vs. Versicolor, it’s [-0.269, 0.338, -
0.701, -0.749], the bias is -0.247, and the penalty weight is 50. 
Evaluation metrics, including recall, F1-score, and accuracy, 
all exceeded 0.93, indicating good model performance. And 
the QUBO model solving time ranged from 0.07 to 0.09 
seconds. 

For model 3, we used the MNIST dataset for image 
classification, applying convolutional neural networks 
(CNNs) with ReLU activation and cross-entropy loss. We 
introduced penalty coefficients for the convolutional and fully 
connected layers, as well as cross-entropy, and minimized the 
loss for image recognition. To prevent overfitting, L2 
regularization was added. The softmax activation function 
was used in the output layer to calculate category probabilities. 

Evaluation metrics, including macro avg and weighted avg 
recall, F1-score, and accuracy, all exceeded 0.98, with a 
support of 10,000, indicating strong model performance. As 
QUBO model solving time was around 211 seconds. 

This paper fully demonstrates the application of the QUBO 
model in optimizing machine learning algorithms, which can 
be extended to other artificial intelligence algorithms and 
represents a future direction for research. 

Acknowledgments 
The authors gratefully acknowledge the financial support 

from 2024 Guangzhou University Student Innovation 
Training Project of Ministry of Education of P.R.C (2024110 
78145). 

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