# A novel feature selection method based on quantum support vector machine

Haiyan Wang<EMAIL_ADDRESS>School of Mathematical and Natural Science,
Arizona State University, Phoenix, AZ 85069, USA

###### Abstract

Feature selection is critical in machine learning to reduce dimensionality and
improve model accuracy and efficiency. The exponential growth in feature space
dimensionality for modern datasets directly results in ambiguous samples and
redundant features, which can severely degrade classification accuracy.
Quantum machine learning offers potential advantages for addressing this
challenge. In this paper, we propose a novel method, quantum support vector
machine feature selection (QSVMF), integrating quantum support vector machines
with multi-objective genetic algorithm. QSVMF optimizes multiple simultaneous
objectives: maximizing classification accuracy, minimizing selected features
and quantum circuit costs, and reducing feature covariance. We apply QSVMF for
feature selection on a breast cancer dataset, comparing the performance of
QSVMF against classical approaches with the selected features. Experimental
results show that QSVMF achieves superior performance. Furthermore, The Pareto
front solutions of QSVMF enable analysis of accuracy versus feature set size
trade-offs, identifying extremely sparse yet accurate feature subsets. We
contextualize the biological relevance of the selected features in terms of
known breast cancer biomarkers. This work highlights the potential of quantum-
based feature selection to enhance machine learning efficiency and performance
on complex real-world data.

feature selection, support vector machine; quantum feature map; genetic
algorithm; breast cancer

##  1 Introduction

Feature selection is a common way to minimize the problem of excessive and
irrelevant features, which is one of the most fundamental problems in machine
learning and pattern recognition. Overcoming the curse of dimensionality is
one of the biggest challenges in building an accurate predictive machine
learning model from high dimensional data. For a feature selection problem
with m features, the objective of feature selection should be to achieve the
highest classification accuracy by finding an optimal feature subset. The
number of possible feature subsets is an optimal solution is NP-hard [45, 46,
49, 51].

Quantum machine learning has emerged as a highly promising application of
quantum technologies. In the search for quantum advantage, the quantum support
vector machines, stand out as a powerful approach, capturing the interest of
diverse research groups [15, 11, 19, 22, 28, 23, 25, 30, 3, 27, 36, 9, 16, 12,
20]. In the work presented by [12], innovative strategies were introduced to
construct quantum kernels for support vector machines (SVM). This method
utilizes classical data and transforms it into the quantum state space through
a quantum feature map, enabling the conversion of the dataset from its
original low-dimensional real space to a higher-dimensional quantum state
space, commonly known as the Hilbert space. More recently, quantum computing
has emerged as a promising platform fort tackling computationally expensive
combinatorial optimization tasks such as feature selection[53, 54, 47, 48].

Grounded in the principles of natural selection, genetic algorithms emerge as
a valuable tool for feature selection [50, 49]. Recently, genetic algorithms
have found application in support vector machines [6]. Genetic algorithms have
been extensively applied in quantum computing [34, 35, 7] and quantum support
vector machines [5, 2, 8]. The concept of using genetic algorithms for the
automatic generation of feature maps in quantum support vector machines was
first introduced by [2]. In addition, [17, 13] proposed algorithms for
automatically generating quantum circuits using multi-objective genetic
algorithms. These algorithms offer a means to optimize gate or circuit
structures, addressing common challenges such as local minima and barren
plateaus [31, 32, 33].

In this paper, we introduce a new feature selection method, quantum support
vector machine feature selection (QSVMF), based on quantum support vector
machine (QSVM) and a multi-objective genetic algorithm (specifically the Non-
dominated Sorting Genetic Algorithm (NSGA-II)). We use five fitness functions
that simultaneously maximizes classification accuracy while minimizing, the
number of selected features, the gate costs of the quantum feature map’s
circuit and the covariance of selected features. The inclusion of the
minimization of quantum gate costs and the feature covariances helps reducing
high computational cost which is a major drawback of using genetic algorithms.
In particular, we separate the fitness functions for local gates and non-local
(controlled-NOT (CNOT)) gates for balanced quantum gates for entanglement,
which complements the previous works [2, 8] where their formulation of fitness
functions has primarily resulted in the suppression of entanglement gates in
the design. We apply QSVMF to the feature selection of the breast cancer
dataset and compare QSVMF with SelectKBest from scikit-learn library and
classical methods. QSVMF outporforms in the most cases.

In addition, the use of NSGA-II in this paper allows us to examine the Pareto
front solutions for an interesting analysis of the trade-off between number of
features and accuracy using the QSVMF method. In particular, we include the
minimization of the total covariance of selected features to reduce the
redundance of features. As a result, we identify the feature subsets with the
minimum number of features for each qubit setting. Despite having very few
features, these compact QSVMF feature subsets can still achieve good accuracy.
This demonstrates another key advantage of the QSVMF approach - its ability to
pare down the feature space to a very sparse set of highly informative
features critical for prediction. The compact high-accuracy feature subsets
could help enable more interpretable and efficient predictive modeling.

Finally, we conceptualize that the features chosen by QSVMF align with
established medically relevant imaging biomarkers for differentiating between
benign and malignant breast lesions. A future comprehensive study on diverse
datasets will fully justify the advantages of QSVMF across various data
distributions and complexities.

Several studies have explored quantum feature selection approaches, including
those based on a quadratic unconstrained binary optimization (QUBO) problem
[53], Hamiltonian encoding and a ground state [54], quantum approximate
optimization algorithm (QAOA) [47] and variational quantum optimization with
black box binary optimization [48]. To the author’s knowledge, this paper
marks the first application of QSVM and NSGA-II for quantum feature selection.

##  2 Feature Selection

Feature selection is a technique aimed at enhancing training efficiency. It
chooses a subset of features for training a model, with the primary objectives
of expediting model training, minimizing memory requirements, and mitigating
overfitting risks. Additionally, since not all features may contribute
significantly to the training process, feature selection plays a pivotal role
in enhancing model interpretability and promoting generalization.

The primary classical feature selection categories can be broadly classified
into three groups: 1) filters; 2) wrappers; and 3) embedded methods, as
discussed in [45]. Filter methods employ statistical techniques to
independently evaluate the relationships between features and the output,
eliminating features that fail to meet predefined thresholds. Wrapper methods
evaluate numerous feature subset models, selecting the most effective features
that maximize overall model accuracy. Embedded methods seamlessly integrate
feature selection into the classifier algorithm. During training, the
classifier adjusts its parameters to determine the importance of each feature
for optimal classification accuracy. Embedded methods strive to strike a
balance between the characteristics of filter and wrapper methods.

In this paper, we propose a new embedded method, QSVMF, for feature selection
based on high-dimensional NSGA-II (Non-dominated Sorting Genetic Algorithm II)
and quantum support vector machine. The new method utilizes a SVM with quantum
kernels to maximize prediction accuracy and NSGA-II to reduce computational
cost by minimizing complexity of quantum circuits and correlation of selected
features. As a result, QSVMF outperforms many classical methods for feature
selection of the breast cancer dataset.

##  3 Quantum Support Vector Machine

###  3.1 Support vector machine

Support vector machines (SVMs) represent robust supervised learning models
employed for classification tasks. The objective is to identify the optimal
hyperplane that effectively maximizes the separation between two classes. In
mathematical terms, considering a labeled training set ${(x_{i},y_{i})}$ where
$x_{i}$ denotes the features and $y_{i}$ signifies the binary labels, this
objective is mathematically formulated as:

$\displaystyle\text{minimize}\quad||w||^{2}$ $\displaystyle\text{subject
to}\quad y_{i}(w\cdot x_{i}+b)\geq 1,\quad\forall i$

Here, $w$ represents the normal vector to the hyperplane, and $b$ is the bias
term. The constraints are in place to guarantee the maximization of the
margin. The points $x_{i}$ closest to the hyperplane are referred to as
support vectors. This primal optimization can be rephrased in the Lagrangian
dual form:

$\displaystyle\max_{\alpha}$ $\displaystyle
L(\alpha)=\sum_{i=1}^{n}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\alpha_{i}\alpha_{j}y_{i}y_{j}x_{i}\cdot
x_{j}$ (1) s.t.
$\displaystyle\sum_{i=1}^{n}\alpha_{i}y_{i}=0,\quad\alpha_{i}\geq 0$

Given a function $\Phi$ that maps $x$ to a higher-dimensional space, this dual
form allows for the utilization of kernels, denoted as
$K(x_{i},x_{j})=\Phi(x_{i})\cdot\Phi(x_{j})$, enabling the implicit evaluation
of inner products. This technique, commonly known as the ”kernel trick,” plays
a crucial role in extending the capabilities of SVMs beyond linear problems.
In essence, kernels facilitate the transformation of data into higher-
dimensional feature spaces, where non-linear problems become linearly
separable.

###  3.2 Quantum kernel method

The Quantum Kernel Estimator described in [12] provides effective strategies
for constructing a quantum kernel for a support vector machine. A major
challenge in the construction of quantum kernel is encoding classical data
into quantum states for quantum computing. [12] achieves this encoding by
mapping data into the quantum state space through a quantum feature map. The
choice of feature map is critical, often dependent on the dataset’s
characteristics to be classified. The feature map’s importance goes beyond
utilizing the quantum state space as a feature space. It also involves how the
data gets mapped into this high-dimensional space.

By employing quantum circuits, a data point
$\mathbf{x}=(x_{j})\in\mathbb{R}^{n}$ undergoes transformation into an n-qubit
quantum feature state, denoted as
$|\Phi(\mathbf{x}\rangle\langle\Phi(\mathbf{x})|$. This naturally gives rise
to the definition of the kernel as
$K(\mathbf{x},\mathbf{z})=|\langle\Phi(\mathbf{x})|\Phi(\mathbf{z})\rangle|^{2}$.
The incorporation of the kernel method into quantum computing seamlessly
occurs through the utilization of a unitary operator
$\mathcal{U}_{\Phi(\mathbf{x})}$ applied to the initial state $|0\rangle^{n}$
as $|\Phi(\mathbf{x})\rangle=\mathcal{U}_{\Phi(\mathbf{x})}|0\rangle^{n}$.
This approach maps data points into the quantum Hilbert space [26], thereby
capitalizing on the significant advantages offered by quantum computing.

$K(\mathbf{x},\mathbf{z})=|\langle\Phi(\mathbf{x})|\Phi(z)\rangle|^{2}=|\langle
0^{n}|\mathcal{U}_{\Phi(\mathbf{x}}^{\dagger})\mathcal{U}_{\Phi(\mathbf{z})}|0^{n}\rangle|^{2}$
(2)

In the work by [12], several well-established feature maps, such as
ZFeatureMap and ZZFeatureMap, are introduced. A customization of quantum
feature maps involves the utilization of more Pauli gates. As an illustration,
a representative form of our customized quantum feature map could assume the
following structure:

$\mathcal{U}_{\Phi(\mathbf{x})}=\left(\exp\left(i\sum_{j,k}\phi_{\\{j,k\\}}(\mathbf{x})Z_{j}\otimes
Z_{k}\right)\exp\left(i\sum_{j}\phi_{\\{j\\}}(\mathbf{x})P_{j}\right)H^{\otimes
n}\right)^{d}$

Here, $P_{j}$ represents one of the rotation gates $I,X,Y,Z$. The quantum
circuit for $d=1$ is depicted in Table 1 for $n=2$ qubits. The coefficient
$\phi_{S}(x)$ is computed, with $S$ denoting a set comprising independent
qubits or qubit pairs, as illustrated below

$\phi_{S}:x\mapsto\begin{cases}x_{i}&\text{if }S=\\{i\\}\\\
2x_{i}x_{j}&\text{if }S=\\{i,j\\}\end{cases}$ (3)

to encode the data.

$\begin{array}[]{c}\Qcircuit<EMAIL_ADDRESS>\lstick{\ket{0}}&\gate{H}&\gate{R}&\targ&\gate{R}&\targ&\qw}\end{array}$ Table
1: Quantum circuit for $n=2$ qubits

The circuit illustrated in Table 1 consists of a layer of Hadamard gates
$H^{\otimes 2}$, followed by an additional layer of single-qubit gates
$R=e^{i\phi_{{j}}(\mathbf{x})P_{j}}$. These $R$ gates are defined by a set of
angles $\phi_{{j}}(\mathbf{x})$, each corresponding to a unique axis, and
$\phi_{{j}}(\mathbf{x})$ is dependent on the feature data. The diagonal
entangling gate $e^{i\phi_{{0,1}}(\mathbf{x})Z_{0}\otimes Z_{1}}$ is
characterized by an angle $\phi_{{0,1}}(\mathbf{x})$ and can be realized with
two CNOT gates and one $R=e^{i\phi_{{0,1}}(\mathbf{x})Z_{1}}$ gate, as
illustrated in Table 1 [12]. In this context, the CNOT gate establishes
entanglement among qubits, specifically acting on two qubits, and is
represented by the matrix:

$\mathrm{CNOT}=\left[\begin{array}[]{llll}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{array}\right]$

The first qubit functions as the control qubit, while the second qubit is
assigned as the target qubit. If the control qubit is in the state
$|0\rangle$, the target qubit maintains its original state. On other hand,
when the control qubit is in the state $|1\rangle$, the target qubit
undertakes a bitwise flip. The classical equivalent of the CNOT gate aligns
with the reversible XOR gate.

Creating a suitable kernel function for diverse datasets poses a significant
challenge. In our investigation, we employ a genetic algorithm to execute a
global optimization strategy for refining the structure of the quantum
circuit. This approach has proven to be effective in overcoming common
challenges, such as barren plateaus, encountered in traditional optimization
methods. Consequently, it produces high-quality learning kernels [2, 10].

##  4 Multi-Objective Genetic Algorithm

Genetic algorithms are optimization methodologies inspired by the principles
of evolution. These algorithms navigate a solution space by iteratively
refining a population of individuals. Across successive generations, genetic
operations guide the selection of offspring with the aim of improving one or
more objectives. This evolutionary process concludes with the identification
of the individuals to optimize fitness functions within a vast configuration
space.

When generating new solutions, the process involves selecting ”parent”
solutions from a predefined pool. Utilizing mechanisms such as crossover and
mutation, a ”child” solution is created, inheriting traits from its ”parents.”
New parent pairs are chosen for each child, and this process is repeated until
a new population of the desired size is formed. The effectiveness and
applicability of a genetic algorithm rely significantly on the choice of
genetic operations. Among these operations, the selection operator is crucial
as it determines a subset of the current population to create a new generation
through crossover and mutation. Mutation involves randomly altering
information in selected individuals, allowing for exploration of distant
regions in the solution space. On the other hand, crossover facilitates more
radical exploration by exchanging genetic information between two individuals.

To guarantee efficiency and convergence, genetic algorithms incorporate early
stopping conditions to evaluate whether the evolutionary process has achieved
its goals. These conditions may include closely examining fitness convergence
or saturation, maintaining a minimum accuracy threshold for continuation, or
defining a maximum generation count. By using these essential components,
genetic algorithms demonstrate their value in tackling complex optimization
problems across various domains

For many problems, an evolutionary multi-objective optimization (EMO) may be
more effective, which involves multiple objective functions that require
either minimization or maximization [1, 21, 4]. Multi-objective optimization,
like single-objective problems, may also have various constraints that all
feasible solutions.

Initialize population Perform Crossover and Mutation Combine Parent and
Offspring Population Evaluate Fitness Functions Non-dominated Sorting and
Rejection Crowding Distance Sorting Check Stopping Criteria Identif Pareto
Fronts Select Sruvivors for Next Generation Apply QSVM yesno Figure 1: Non-
dominated Sorting Genetic Algorithm (NSGA-II)

Optimal solutions in multi-objective optimization are described by the
mathematical concept of partial ordering, expressed through the term
”domination.” This paper only addresses unconstrained optimization problems,
which do not involve any equality, inequality, or bound constraints. We say
that solution $\mathbf{x}^{(1)}$ dominates another solution,
$\mathbf{x}^{(2)}$, only if both of the following conditions hold true:

  1. 1.

The solution $\mathbf{x}^{(1)}$ is no worse than $\mathbf{x}^{(2)}$ in all
objectives (it holds for each component of the vector.).

  2. 2.

The solution $\mathbf{x}^{(1)}$ is strictly better than $\mathbf{x}^{(2)}$ in
at least one objective ( it holds for at least one component of the vector)

Points that persist without being dominated by any other members within a set
are labeled as non-dominated points. A notable characteristic of such points
is their trade-off relationship, where improvements in one objective affect
trade-offs in at least one other objective. This trade-off property observed
among non-dominated points is interesting as it promotes the exploration of a
diverse array of such points before arriving at a final decision. The Pareto
front represents the collection of such non-dominated solutions, forming a
frontier where any improvement in one objective necessitates a trade-off in
another.

The Non-dominated Sorting Genetic Algorithm (NSGA-II) emerged as a response to
the deficiencies observed in early evolutionary algorithms, particularly the
absence of elitism, as discussed in [4]. It has become a widely used technique
in the field of Evolutionary multi-objective optimization.

As we structure our fitness functions to encompass the minimization of gate
costs and the total covariances of selected features, the Pareto fronts
derived for QSVMF provide insights into the strategy of balancing high
accuracy with minimal gate costs and selected features. This facilitates the
analysis of cost-effective trade-off strategies. A more thorough investigation
into the multiple-dimensional Pareto fronts and their projections onto low-
dimensional spaces could offer a deeper understanding of the significant
impact of quantum feature maps on feature selection.

##  5 Quantum Kernel for Support Vector Machine

###  5.1 Genetic quantum feature map

Utilizing the approach outlined in [2] to generate quantum feature maps
through a multi-objective optimization process, this paper employs the NSGA-II
algorithm to automatically generate quantum feature maps for support vector
machines. As previously mentioned, the NSGA-II algorithm incorporates elitism
and other features to ensure diversity within the Pareto set. It searches for
circuits that, when trained using a support vector machine, maximize accuracy
while simultaneously minimizing circuit complexity and feature redundancy.

Gate costs are quantified in terms of local gates, including Hadamard gates
and rotation gates, as well as CNOT gates. In this study, we introduce a
refined approach by separate CNOT gates from local gates and incorporating
additional optimization functions into the circuit generation process.
Building upon the encoding method established in [2, 8], our modifications aim
to create a quantum circuit with minimal gate costs and covariances of
features while maximizing accuracy. Assume that the number of qubits is $N$.
The encoding scheme starts with the number of bits equal to the number of
features to indicating if a feature is selected, and then $N$ Hadamard gates,
followed by $N$ bits indicating whether if a rotation is applied to each
qubit, and $\binom{N}{2}$ bits for the number of potential CNOT gates and an
additional two bits to denote rotations with respect to $X$, $Y$, or $Z$. The
last two bits are for repetition information. The size of the encoding bit
string is given by

$\text{number of features}+N+\binom{N}{2}+4$

Such scheme includes possible Hadamard entanglement, achievable through two
rotation gates and a CNOT gate, as illustrated in Table 1. We incorporate the
input data into the encoding. If the feature number is the same as the number
of qubits, ensuring that the $i$-th gene operating on the qubits of the
quantum register depends on the $i$-th variable from the input data
$\mathbf{x}\in\mathbb{R}^{n}$, as expressed in Equation (3). If the feature
number is larger than the number of qubits, we randomly select the number of
feature as the number of qubits for embedding the feature data to the feature
maps. If the feature number is less than the number of qubits, we randomly
copy some features to match the number of qubits for embedding the feature
data to the feature maps.

The genetic algorithm initiates the population randomly, and each individual
will associate and quantum circuit and undergoes fitness evaluation. Utilizing
the training dataset, the circuit assesses kernel matrices, which are
integrated into SVM algorithms to compute the fitness function. Individuals
with superior fitness are more likely to undergo genetic operations, including
mutation, crossover, and selection, leading to the creation of a new
generation of individuals or quantum circuits. This iterative process
continues until convergence criteria, including comparison of accuracy and the
number of generation, are met. To illustrate the our encoding scheme, we
consider a dataset with four features, three qubits with 14 bits per
individual in the population, as depicted in Figure 2. Figure 3 is the
corresponding circuits of the quantum feature map.

10111111001110 Each bit indicates if a feature is selected Each bit indicates
if there is a rotation for each qubit two bits for rotation with respect to
$X,Y,Z$ three bits for entanglements two bits for repts Figure 2: Example for
the encoding scheme for 4 features and 3 qubits: the binary bits are
[1,0,1,1,1, 1, 1, 1, 0, 0, 1, 1, 1, 0], the first four bits indicate that
features 0,2,3 are chosen, the next three bits for 3 rotation gates, and next
two bits indicates the rotation with respect to $X,Y,Z$, followed 3 bits for
possible entanglements, the last two bits are for repetitions. Figure 3:
Quantum circuits for Figure 2

###  5.2 Fitness functions

fitness functions are the he objective or cost function for measuring the
effectiveness of QSVMF, taking a solution as input and providing its fitness
score. Introducing five fitness functions inspired by [2] below (4), our goal
is two-fold: to maximize accuracy and minimize gate costs and covariance of
selected features. Recognizing the distinct costs associated with different
gates [2, 8, 18], we differentiate CNOT gates from local gates, leading to the
introduction of two fitness optimization functions for gate costs. Gate costs
are computed by adding the number of gates used in the operations required for
quantum computer implementation. We choose not to integrate weights into the
objective functions, as the introduction of such weights might have a
potential impact on the optimization process in a manner that is intentionally
undesirable. In our pursuit of this optimization, we conduct a evaluation
using K-fold cross-validation. Here accuracy represents the mean accuracy
achieved on the provided test data and labels over the k-fold cross-
validation.

$\left\\{\begin{array}[]{lr}\text{(maximize)}\text{Fitness 1}=\text{prediction
accuracy}\\\ \text{(minimize)}\text{Fitness 2 }=\text{ local
gates}=\text{Rgate}+\text{Hgate}\\\ \text{(minimize)}\text{Fitness 3
}=\text{CNOT gate}\\\ \text{(minimize)}\text{Fitness 4 }=\text{feature
count}\\\ \text{(minimize)}\text{Fitness 5 }=\text{feature
variance}\end{array}\right.$ (4)

##  6 Feature Selection with QSVMF

###  6.1 Breast cancer dataset

Our analysis uses the breast cancer dataset retrieved from Sklearn [29, 37].
This dataset encompasses measurements of breast tissue obtained through
medical imaging techniques, featuring various metrics. The primary goal is to
determine the nature of a tumor, distinguishing between benign (non-cancerous)
and malignant (cancerous and hazardous) tumors. The breast cancer dataset from
scikit-learn, consists of 569 samples, incorporating 30 real, positive
features. These features include attributes related to cancer masses,
including mean radius, mean texture, mean perimeter, and more. Among these
samples, 212 instances are labeled as ”malignant,” while 357 instances are
labeled as ”benign.” We opt not to employ principal component analysis (PCA)
for dimension reduction for intensifying the challenge for achieving high
accuracy.

###  6.2 QSVMF Procedure

The experiment utilizes high-dimensional NSGA-II (Non-dominated Sorting
Genetic Algorithm II) and quantum feature mapping to enhance accuracy. After
encoding each individual in the population for quantum feature maps, genetic
algorithm operators such as selection, crossover, and mutation are applied to
generate the subsequent generation of individuals. Subsequently, we calculate
the kernel of a quantum SVM while training a classifier with the provided
training dataset. In each iteration, we assess the classifier’s accuracy using
the test dataset and evaluate the effective size of the circuit in terms of
the number of local and entangling gates it incorporates. Special precautions
are taken to prevent overfitting by implementing K-fold cross-validation and
comparing the performance with the baseline model. The procedure is structured
into a sequence of interconnected steps depicted by distinct rectangular
blocks, as illustrated in Figure 4, and can be further described as follows.

Data preprocess (K-fold, population encoding,..) NSGA-II process (Crossover,
Mutation) Feature selection according to encoding scheme Compute quantum
kernel Compute accuracy using QSVM NSGA-II process (sorting and refining)
K-fold criteria NSGA-II repetition Extract selected features from Pareto
fronts Perform predictions for comparison noyes Figure 4: Experimental
procedure with NSGA-II and quantum feature map

  * •

Data Preprocess: The first step comprises data preprocessing utilizing a
K-fold strategy, which divides the dataset into subsets for both training and
testing. Additionally, individuals within populations are encoded for quantum
feature maps

  * •

NSGA-II Process (Crossover, Mutation): It encompasses various genetic
operators such as crossover and mutation, designed to evolve a population of
potential solutions to improve their quality with respect to multiple
objectives.

  * •

Feature selection According to Encoding Scheme: Features are selected Before
computing kernel matrix according to the encoding scheme.

  * •

Construct Quantum Feature Map and Compute Kernel Matrix: We construct a
quantum feature map, a pivotal component of SVM that transforms classical data
into a quantum state. The process also includes the computation of a kernel
matrix to capture data relationships.

  * •

Classical SVM (quantum kernel) to Compute Accuracy: A classical SVM is
utilized with a quantum kernel to calculate the accuracy of the model’s
predictions, enhancing the efficacy of this potent classical method in machine
learning.

  * •

NSGA-II Process (Sorting and refining): The individuals are sorted based on
their dominance, potentially deleting and refining the solutions generated
earlier.

  * •

Extract Pareto Fronts and the most frequent features: The final step aims to
obtain $K$ Pareto fronts and the most $m$ frequnt features from the resulting
Pareto fronts associated with the highest accuracy, where $m$ is the average
of the numbers of features of each K-fold cross validation.

  * •

Perform predictions with both classical and quantum classifiers for
comparison: With the selected feature list, we perform various predictions
with both classical and quantum classifiers for comparison.

With our training and testing datasets prepared, we follow the previously
outlined procedure to compute the training and testing kernel matrices. We
initiate the process with an initial random population of 100 individuals,
represented as binary string. Each string comprises $\text{number of
features}+N+\binom{N}{2}+4$ bits, where $N$ denotes the number of qubits.
Employing a 20% crossover and mutation rate, we aim for a more realistic
accuracy by utilizing the 5-fold cross-validation method, resulting in five
distinct training and test subsets. Subsequently, we average accuracy and
other outputs for a comprehensive assessment. We employ a combination of
generation number and accuracy as a stopping criterion. These quantum kernels
can be seamlessly integrated into classical kernel methods. Notably, NSGA-II
typically runs around 100 generations to achieve higher accuracy compared to
classical methods.

1

Data: training data X, labels y, number of qubits

Result: selected features, comparsions

2 while  _k-lold validation not done_ do

3 while  _NSGA-II repetition not done_ do

4 initialize population; for  _each generation_ do

5 Crossover and Mutation;

6 Construct Quantum Feature Map and Compute Kernel Matrix;

7 Use classical SVM with quantum kernel to compute accuracy and others;

8 update fitness;

9 Sorting individuals;

10 Rejection and Crowding Distance Sorting;

11 end for

12

13 end while

14

15 end while

16Obtain Pareto front solutions; compute accuracy with 5-fold cross
validation; get the most $m$ frequent selected features from the $k$ Pareto
fronts associated with the highest accuracy, where $m$ is the average of the
numbers of features;

17 retrain final QSVM for comparison with other classical classifiers.

return selected features; comparisons

Algorithm 1 QSVMF pseudo code

This multi-stage procedure is designed to achieve optimal solutions for
complex optimization tasks while leveraging quantum computing principles and
features an integrated approach that combines classical and quantum
techniques. It is a sophisticated and comprehensive methodology for addressing
challenging optimization and feature selection problems.

###  6.3 Comparisons of QSVMF and SelectKBest

SelectKBest is a feature selection method provided by the scikit-learn library
in Python. The SelectKBest function takes two main arguments: score-func,
which specifies the scoring function to use, and k, which specifies the number
of top features to select. It selects the k highest scoring features based on
a scoring function. In this paper, the chi2 and f-regression scoring functions
from scikit-learn were used for SelectKBest. The number $k$ of features
selected by SelectKBest was set to be the same as the number of features
selected by the QSVMF method. This allows a direct comparison between the
classical SelectKBest and the quantum-based QSVMF in terms of their selected
feature subsets. SelectKBest provides a simple baseline for feature selection
that can be useful for benchmarking more advanced techniques like QSVMF.

Feature Selection Method qubits 2 3 4 5 6 7 8 9 10 QSVMF [13,23] [20,22]
[3,13,21] [13,17,22,29] [11,20,21,22] [1,13,18,22,29] [18,21,22,24]
[1,11,14,20,24] [0,1,11,12,20,24,25] SelectKBest (chi2) [3, 23] [3, 23] [3,
13, 23] [3, 13, 22, 23] [3, 13, 22, 23] [2, 3, 13, 22, 23] [3, 13, 22, 23] [2,
3, 13, 22, 23] [0, 2, 3, 13, 20, 22, 23] SelectKBest (f-regression) [22, 27]
[22, 27] [7, 22, 27] [7, 20, 22, 27] [7, 20, 22, 27] [2, 7, 20, 22, 27] [7,
20, 22, 27] [2, 7, 20, 22, 27] [0, 2, 7, 20, 22, 23, 27]

Table 2: Features selected by QSVMF and SelectKBest

Table 2 compares the feature subsets selected by QSVMF and the classical
SelectKBest method with chi2 and f-regression scoring functions. The number of
qubits used for QSVMF ranges from 2 to 10, resulting in feature subsets of
increasing sizes. SelectKBest was configured to select the same number of
features as QSVMF for each qubit setting, to enable a direct comparison. As a
result, although SelectKBest does not directly depend on qubits, because the
choice of $k$ may depend on the number of qubits. As shown in the table, while
there is some overlap, QSVMF and SelectKBest generally select different
feature subsets. Even for SelectKBest, chi2 and f-regression also select
different subset of features. SelectKBest with Chi2 operates by evaluating the
statistical significance of each feature independently with respect to the
target variable and does not take into consideration the feature interactions.
In contrast, QSVMF evaluates the perform of overall performs of the
classifiers. As a result, selections vary more with the number of qubits.

Table 3 compares the predictive accuracy of QSVMF against the performance of
the SelectKBest with chi2 based on several SVMs with classical kernels
(Linear, Poly, RBF, Sigmoid) and Logistic Regression, Naive Bayes K-Neighbors
and Decision Trees. For each model, SelectKBest uses the same number of
features as QSVMF for a fair comparison across different qubit settings. As
shown in the table, QSVMF achieved higher accuracy than all the classical
models for nearly all qubit counts. The accuracy advantage of QSVMF over the
classical models was most pronounced with fewer qubits, with gaps of up to
0.05. However, QSVMF maintained an accuracy edge even with higher qubit
counts. The classical models trended toward slightly higher accuracy with more
qubits, as the feature space expanded. But none could surpass the QSVMF
accuracy. This indicates QSVMF is able to identify more informative features
for prediction compared to SelectKBest, leading to higher accuracy models. The
results highlight the potential of quantum-based feature selection to extract
better feature subsets and improve predictive performance compared to
classical methods.

Prediction Method | qubits
---|---
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
QSVMF | 0.95 | 0.95 | 0.94 | 0.97 | 0.97 | 0.94 | 0.96 | 0.98 | 0.97
SVM (Linear) | 0.90 | 0.90 | 0.90 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.93
SVM (Poly) | 0.91 | 0.91 | 0.92 | 0.93 | 0.93 | 0.93 | 0.93 | 0.93 | 0.92
SVM (RBF) | 0.92 | 0.92 | 0.91 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92
SVM (Sigmoid) | 0.82 | 0.82 | 0.80 | 0.83 | 0.83 | 0.85 | 0.83 | 0.85 | 0.87
Logistic Regression | 0.90 | 0.90 | 0.90 | 0.92 | 0.92 | 0.93 | 0.92 | 0.92 | 0.92
Naive Bayes | 0.91 | 0.91 | 0.91 | 0.91 | 0.91 | 0.91 | 0.91 | 0.91 | 0.91
K-Neighbors | 0.91 | 0.91 | 0.91 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 | 0.93
Decision Tree | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92 | 0.92
Table 3: Accuracy comparison for prediction methods: QSVMF vs SelectKBest ( with chi2 and the same number of features as QSVMF). Prediction Method | qubits
---|---
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
QSVMF | 0.95 | 0.95 | 0.94 | 0.97 | 0.97 | 0.94 | 0.96 | 0.98 | 0.97
SVM (Linear) | 0.93 | 0.93 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95
SVM (Poly) | 0.94 | 0.94 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95
SVM (RBF) | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.94 | 0.95 | 0.94 | 0.94
SVM (Sigmoid) | 0.94 | 0.94 | 0.94 | 0.94 | 0.93 | 0.94 | 0.93 | 0.95 | 0.95
Logistic Regression | 0.93 | 0.93 | 0.92 | 0.93 | 0.93 | 0.95 | 0.93 | 0.95 | 0.95
Naive Bayes | 0.95 | 0.95 | 0.95 | 0.94 | 0.95 | 0.94 | 0.94 | 0.94 | 0.94
K-Neighbors | 0.93 | 0.93 | 0.93 | 0.93 | 0.93 | 0.95 | 0.93 | 0.95 | 0.95
Decision Tree | 0.93 | 0.93 | 0.93 | 0.93 | 0.93 | 0.95 | 0.93 | 0.95 | 0.95
Table 4: Accuracy comparison for prediction methods: QSVMF vs SelectKBest
(with f-regression and the same number of features as QSVMF).

Table 4 compares the predictive accuracy of QSVMF against the performance of
the SelectKBest with f-regression based on several SVMs with classical kernels
(Linear, Poly, RBF, Sigmoid) and Logistic Regression, Naive Bayes K-Neighbors
and Decision Trees. For each model, SelectKBest uses the same number of
features as QSVMF for a fair comparison across different qubit settings. Table
4 shows an interesting trend in the accuracy comparison between QSVMF and
SelectKBest with f-regression scoring. For small qubit counts of 2-4,
SelectKBest actually matches or slightly exceeds the accuracy of QSVMF.
However, as the number of qubits increases, QSVMF significantly outperforms
SelectKBest, with accuracy advantages of up to 0.06 for 9-10 qubits. This
suggests QSVMF is better able to leverage the information from larger feature
spaces to identify predictive feature subsets. In contrast, the performance of
SelectKBest peaks with fewer features and then degrades as more irrelevant
features are included. Comparing these results to the previous table using
SelectKBest with chi2 scoring, we see a similar overall trend of QSVMF
achieving higher accuracy than the classical method. However, the margin of
improvement is smaller here.

Overall, across both tables 3,4 using different SelectKBest settings, QSVMF
demonstrates a robust accuracy advantage, affirming the potential of quantum-
based feature selection to extract superior feature subsets compared to
classical methods. The degree of improvement depends on the dataset
characteristics and classical benchmark method. But QSVMF exhibits a
consistent capability to identify more predictive features leading to higher
accuracy models.

###  6.4 Minimal features in QSVMF

Table 5 shows an interesting analysis of the trade-off between number of
features and accuracy using the QSVMF method. By examining the Pareto front
solutions, the table identifies the feature subsets with the minimum number of
features for each qubit setting. Despite having very few features, these
compact QSVMF feature subsets can still achieve good accuracy, often over 80%.
For example, with just a single feature (feature 13), QSVMF attains 89%
accuracy for 7 qubits. With 2 features for 3 qubits, accuracy reaches 87%.
Even with 7-10 qubits, QSVMF can identify tiny feature subsets of 1-3 features
that provide accuracy exceeding 75%. This demonstrates a key advantage of the
QSVMF approach - its ability to pare down the feature space to a very sparse
set of highly informative features critical for prediction. The fact that
QSVMF can maintain such high accuracy with so few features highlights its
effectiveness at zeroing in on the most relevant variables and eliminating
redundant or irrelevant ones. The table provides an estimate of the minimum
number of features needed to capture the predictive signal in the data using
QSVMF. The compact high-accuracy feature subsets could help enable more
interpretable and efficient predictive modeling.

qubits 2 3 4 5 6 7 8 9 10 Minimal Features [13] [20,22] [13] [28] [14] [7]
[18,23] [1,14,18] [21] Accuracy 0.75 0.87 0.82 0.69 0.74 0.89 0.86 0.83 0.76

Table 5: Minimal features in Pareto fronts and their accuracy by QSVMF

###  6.5 Combined predictive capability

The chi-squared test used in SelectKBest evaluates each feature independently
for correlation with the target, while QSVMF considers the combined predictive
capability of the feature subsets. This likely explains some of the difference
in features selected by the two methods. [43] showed that individually
predictive features do not necessarily combine to give the best classification
performance. They recommend multivariate feature selection over univariate
filtering. [51] note univariate filters assume independence between features,
while many datasets have dependent features. Multivariate methods are better
suited in such cases.

The key point remains that univariate filters like chi2 do not account for
inter-feature dependencies and redundancy, while QSVMF and other multivariate
methods provide a more overall perspective for feature selection. In
particular, QSVMF include a fitness function to reduce the redundance by
minimizing the covariance of the selected features. This likely explains some
of the performance gap observed between them.

###  6.6 Local and Non-local gates in QSVMF

At the last, Table 6 provides valuable insights into the average quantities of
local and CNOT gates associated with our quantum circuits. This table shows
the configuration of the quantum gates for the best accuracy. It’s noteworthy
that in our experiments with the dataset, we observe a reasonable
proportionality between the number of CNOT gates and the number of local
gates. This finding along with [52] deviates from the observations made in [2,
8], where entanglement gates were predominantly suppressed. We separate the
fitness functions for local gates and non-local (controlled-NOT (CNOT) )
gates. By doing so, we obtain a set of non-dominated points that reveal a more
balanced configuration of quantum circuits for the quantum kernel map,
incorporating a proportional number of CNOT gates to facilitate entanglement.
This result complements the previous works [2, 8] where their design of
fitness functions has predominantly led to a suppression of entanglement
gates. In addition, the experiments in this paper scale well for larger qubits
than the previous work in [14] where their experiment requires a significantly
larger amount of computational resources.

Table 6: Gate numbers of Quantum feature map for the best accuracy of QSVMF. Gate | qubits
---|---
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
Local gates | 3.8 | 4.8 | 6.4 | 7.8 | 8.6 | 11.4 | 13.6 | 15.2 | 18
CNOT gates | 1.2 | 0.8 | 2 | 4 | 2.4 | 4.8 | 6 | 9.2 | 11.6

##  7 Interpretation of feature selection results

###  7.1 Feature selections in the context of breast cancer

The table 7 displays the index versus feature names for the breast cancer
dataset. The left column represents the indices, and the corresponding right
column shows the feature names. This format provides a clear reference for
understanding the order and names of features in the dataset.

Index | Feature Name
---|---
0 | mean radius
1 | mean texture
2 | mean perimeter
3 | mean area
4 | mean smoothness
5 | mean compactness
6 | mean concavity
7 | mean concave points
8 | mean symmetry
9 | mean fractal dimension
10 | radius error
11 | texture error
12 | perimeter error
13 | area error
14 | smoothness error
15 | compactness error
16 | concavity error
17 | concave points error
18 | symmetry error
19 | fractal dimension error
20 | worst radius
21 | worst texture
22 | worst perimeter
23 | worst area
24 | worst smoothness
25 | worst compactness
26 | worst concavity
27 | worst concave points
28 | worst symmetry
29 | worst fractal dimension
Table 7: Index versus Feature Names for Breast Cancer Dataset

As indicated in Table 2, for 2 qubits, QSVMF selects the area error (index 13)
and worst radius (index 23) features. This indicates these two features
related to the radius and area measurements contain the most predictive signal
for the target variable. With 3 qubits, QSVMF selects the worst compactness
(index 20) and worst perimeter (index 22). These features capture deviations
in shape compactness and perimeter which are informative for prediction. For 4
qubits, mean area (index 3) is added, suggesting average area provides
additional discriminative information beyond the shape irregularity features.
As we move to 5 qubits and beyond, QSVMF begins selecting features related to
other aspects of shape like smoothness and symmetry. The worst symmetry (index
29) and mean smoothness error (index 14) indicate predictive information in
irregularity of the symmetry and smoothness.

Overall, QSVMF prioritizes features measuring shape deformations,
irregularities, and extremes, rather than average values. This includes area,
perimeter, compactness, smoothness and symmetry deviations. This provides
insight into what aspects of the shape characteristics are most informative
for predicting the diagnostic target variable using the quantum feature
selection approach.

The focus on area and perimeter-related features makes sense from a medical
perspective, as cancerous tumors often exhibit irregularities in size and
shape boundaries compared to benign masses. The area and perimeter errors and
extremes capture deviations that can differentiate between malignant and
benign growths. The study found that missed breast cancer tumors on mammograms
tend to be smaller, rounder, and have more circumscribed margins compared to
correctly identified tumors [38]. This supports the selection of area,
perimeter and compactness related features. [39] reports that asymmetry is
more common in malignant than benign breast lesions. This provides evidence
for the relevance of the worst symmetry feature. [40] reports that malignant
lesions have rough, irregular surface contours while benign lesions have
smooth, regular surfaces. This backs the selection of the smoothness error
feature, shape analysis of breast cancer on mammograms using digital image
processing.

Overall, these studies demonstrate that the features selected by the
unsupervised QSVMF align with medically-relevant morphological indicators used
by radiologists to evaluate malignancy on breast cancer imaging.

###  7.2 Minimal feature sets in the context of breast cancer

A key reason these minimal QSVMF feature sets maintain high predictive
performance is because they capture the most informative traits that
differentiate between the classes (in this case benign vs malignant). As the
references below demonstrate, often a small number of morphological features
contain most of the discriminative signal for breast cancer diagnosis on
imaging. For example, [41] found that features quantifying shape irregularity,
spiculation and margin characteristics were the most useful for
differentiating between malignant and benign masses on mammography. So even
2-3 features encoding these shape attributes can attain high accuracy.
Similarly, [42] showed textural features pertaining to homogeneity and
contrast were highly effective at malignancy prediction from ultrasound
images. This aligns with QSVMF prioritizing texture and morphology measures.

By removing redundant features, QSVMF enables more efficient and interpretable
models without sacrificing predictive power. Overall, studies have shown that
a few careful selected features capturing variations in shape, texture and
boundary can classify breast lesions with up to high accuracy. The quantum
algorithm is likely identifying these most informative traits for
discrimination.

##  8 Conclusion and discussion

In this work, we propose a new feature selection method called QSVMF that
integrates quantum support vector machines (QSVMs) with (NSGA-II). The key
innovation of QSVMF is the use of five simultaneous fitness objectives during
optimization: 1) maximizing classification accuracy on the training data, 2)
minimizing quantum gate costs for the feature map circuit, 3) minimizing
covariance between selected features, 4) local quantum gates and 5) non-local
CNOT gates. The inclusion of quantum cost and covariance minimization helps
reduce the substantial computational expenses typical of genetic algorithms.
We apply QSVMF for feature selection on a breast cancer dataset and compare
its performance against SelectKBest from scikit-learn and other classical
approaches. Our results demonstrate QSVMF achieves superior feature subsets
compared to these alternatives, showcasing the benefits of our multi-objective
quantum-aware fitness functions. This highlights the potential of QSVMF as an
efficient, effective feature selection technique for complex datasets. The
feature selections made by QSVMF align consistently with established
distinctions in the characteristics of malignant and benign masses. This
underscores the potential of QSVMF in facilitating meaningful feature
selection for complex data.

The design of the five fitness functions in this work for the NSGA-II
component within QSVMF offers an additional advantage by generating Pareto
optimal solutions. This feature allows for an exploration of the trade-off
between minimizing the size of the feature set and maximizing accuracy. An
examination of the Pareto front can pinpoint the most concise feature subsets
produced by QSVMF that still exhibit reasonable accuracy for each qubit
setting. This sparsity underscores a significant strength of the QSVMF
approach – its capacity to significantly reduce the feature space while
retaining predictive capability. These compact and accurate feature subsets
have the potential to enhance the interpretability and efficiency of
predictive modeling.

The experimental outcomes presented in this study reveal the inclusion of
proportional CNOT gates for the promotion of entanglement. These observations
offer additional information to previous studies in [2, 8], where the
influence of CNOT gates was notably diminished. The examination of high-
dimensional non-dominated points provides valuable insights into the impact of
entanglement gates on the effectiveness of quantum support vector machines.
The search for effective design of entanglement gates in quantum computing is
an important research area [14].

The potential advantages of using quantum kernel estimation to enhance machine
learning models are an active area of research [36]. There is no universally
optimal approach, as model performance depends on the distribution and
characteristics of the training data. Various metrics have been proposed to
evaluate the efficiency of quantum kernels, such as quantifying the geometric
difference of embedded data [9], which provides insights into prediction error
patterns [16]. In this work, we applied the proposed QSVMF method to a breast
cancer dataset. While results were promising, further analysis on diverse
datasets is needed to fully characterize when quantum-enabled feature
selection can outperform classical alternatives. This requires additional
studies how factors like training data distribution, feature space structure,
and dataset complexity influence the benefits of our quantum kernel approach
over classical kernels. Ongoing work is investigating these relationships
between data properties and quantum advantage.

##  9 Declarations

Availability of data and materials: The datasets used in paper are publicly
available in the Python package sklearn. The code will be available upon
request.

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