# A preprocessing perspective for quantum machine learning classification
advantage using NISQ algorithms

Javier Mancilla
Stafford Computing LLC,
16192 Coastal Highway, Lewes,
DE 19958, Delaware, USA
<EMAIL_ADDRESS>
Christophe Pere
INTRIQ,
Department of Computer Science
and Software Engineering,
Université Laval Québec, Canada
<EMAIL_ADDRESS>

###### Abstract

Quantum Machine Learning (QML) hasn’t yet demonstrated extensively and clearly
its advantages compared to the classical machine learning approach. So far,
there are only specific cases where some quantum-inspired techniques have
achieved small incremental advantages, and a few experimental cases in hybrid
quantum computing are promising considering a mid-term future (not taking into
account the achievements purely associated with optimization using quantum-
classical algorithms). The current quantum computers are noisy and have few
qubits to test, making it difficult to demonstrate the current and potential
quantum advantage of QML methods. This study shows that we can achieve better
classical encoding and performance of quantum classifiers by using Linear
Discriminant Analysis (LDA) during the data preprocessing step. As a result,
Variational Quantum Algorithm (VQA) shows a gain of performance in balanced
accuracy with the LDA technique and outperforms baseline classical
classifiers.

_K_ eywords Quantum Machine Learning $\cdot$ Quantum data encoding $\cdot$
Classical encoding $\cdot$ Dimensionality reduction

##  1 Introduction

Machine Learning (ML) is a predominant tool nowadays to solve several
challenges in different industries, such as credit scoring Provenzano et al.
(2020), fraud analysis Tiwari et al. (2021), product recommendation Rohde et
al. (2018), and demand forecasting Masini et al. (2020), among other
extensively explored use cases. Under this premise, the research of the
quantum computing properties applied to ML has expanded rapidly in recent
years since a proven advantage could be a highly useful cross-industry.

The recent progress of these explorations in Quantum Machine Learning (QML)
Mishra et al. (2021) put a spotlight on quantum technology, introducing a
challenge to determine if QML will provide an advantage over classical machine
learning or not. The actual devices are noisy, meaning that the depth or
consecutive gate operations are limited (Ristè et al., 2013; Burnett et al.,
2019; Wang et al., 2021). Qubits will lose their entanglement and so, the
information. These devices make up the NISQ era Preskill (2018) and limit the
use of quantum algorithms or hybrid algorithms to be useful Callison and
Chancellor (2022).

A few cases are already in the market showing promising results and some
companies’ commitment to the quantum machine learning journey. One example is
CaixaBank (Spanish Bank) which is working and testing QML models using the
Pennylane quantum framework to define a scoring model for risk assessment
CaixaBank (2022).

One of the major concerns to even grasp a reliable result remains on the
input/output concept and on the number of available good qubits to use. The
bound of 100+ was reached by IBM Chow et al. (2021), but it is still
insufficient to use complex algorithms that require thousands or millions of
qubits depending on the type of problem to be addressed Dalzell et al. (2020).

To be practical in a business context, QML techniques need to avoid the small
number of qubits constraint and create a methodology to use big datasets on
the current NISQ devices. Previous works showed great potential in splitting
big circuits and learning the weights of the different gates separately, or
reusable qubits for image classification Haug et al. (2021).

In this paper, we approach the input problem by comparing different
preprocessing methods and classifier methods on small and larger datasets with
a binary target. The objective is to determine a specific architecture for
preprocessing, dimensionality reduction of the dataset structure, the encoding
manner and the corresponding classifier. We demonstrate that using Linear
Discriminant Analysis (LDA) within the preprocessing phase is better than
Principal Component Analysis (PCA) when the dataset possesses an important
number of features. We generalize this approach by studying the effect of LDA
on the encoding qubits.

Different tabular datasets (Section 2) are used to understand the link between
the number of features and the encoding process (Section 5). We develop a
pipeline (Section 3) to compare the classical and quantum classifier. This
study leads us with a review of the current literature to determine and
discuss rules to obtain a quantum advantage with the current NISQ devices.

##  2 Datasets

The dataset selection in this research aims to emulate real business cases
scenario where the users can find imbalanced dataframes, a small or large
number of features and also represent - in this case - the financial behavior
of a group of people. We used well-known datasets extracted from UCI and
Kaggle to achieve more than 100 features per datapoint in a CSV file for one
of the cases. Nevertheless, we wanted to use an even larger dataset, but it
was extremely difficult to find information with more variables and features
in a public and open license manner.

###  2.1 UCI - Default of Credit Card Clients Dataset

The dataset111https://www.kaggle.com/datasets/uciml/default-of-credit-card-
clients-dataset contains information about credit card clients collected in
Taiwan from April 2005 to September 2005 (Yeh and Lien, 2009). The data
possesses 25 features and 30,000 rows corresponding to individual clients.
This data was extracted from Kaggle but is originally stored in the well-known
University of California, Irvine (UCI) dataset repository.

The objective of using this dataset is to deal with a classification problem
assessing the prediction of default under credit card usage. This data is
imbalanced, having 22% of defaulters and 78% of non-defaulters. The variables
have demographics, payments, billing and current credit card information
features.

In this research, the dataset was used as it is and without modifications such
as oversampling, undersampling, SMOTE or any previous transformation until we
applied the preprocessing designed for the quantum pipeline.

###  2.2 Fraud Detection Dataset

This dataset222https://www.kaggle.com/datasets/volodymyrgavrysh/fraud-
detection-bank-dataset-20k-records-binary reflects information of a bank fraud
transactions on 20,468 datapoints and 114 features. This data was extracted
from Kaggle and was originally uploaded by Volodymyr
Gavrish333https://www.kaggle.com/volodymyrgavrysh.

The objective of using this dataset is to identify which users are fraudsters
or not due to the transactional information. The data is imbalanced with 27%
under class 1 and 73% in class 0.

In this research, the dataset was used as it is as well and without
modifications until we apply the preprocessing designed for the quantum
pipeline.

An important point to be highlighted in this dataset is that looks very
similar to a real-world scenario situation (+100 features and 1,000s of data
points), but we didn’t manage to confirm the source of the information so the
assumption is that must be synthetic or generated. Anyway, this file has more
than 2,000 downloads and 19,000 views on Kaggle and remains one of the most
used dataframes to explore classification techniques.

##  3 Methods

###  3.1 Dimensionality reduction

PCA is one of the predominant structures for dimensionality reduction in the
exploration of classic data into QML algorithms (Mensa et al., 2022). This
technique is used to reduce the features and compress them into N variables to
match a set of N qubits available to run a classification algorithm using a
gate-based quantum circuit. This method is commonly used for unsupervised
linear transformation and to find the maximum variance in high-dimensional
data. PCA reduces dimensions by examining the correlation between various
features, creating orthogonal axes, or principal components, with the
direction of maximum variance as a new subspace.

There are many alternatives to PCA, but one of them demonstrates a significant
impact when we are dealing with quantum classification problems. LDA is a
supervised method that considers class labels by reducing the number of
dimensions. LDA seeks to identify a subspace of features that optimizes class
separability optimally. LDA operates by computing the d-dimensional mean
vector for each class label and then constructing a scatter matrix within each
class and between them.

As we mentioned, both PCA and LDA are linear transformation techniques that
decompose matrices into eigenvalues and eigenvectors. PCA is unsupervised and
does not consider class labels, whereas LDA does. These two techniques will be
applied to classical data preprocessing for QML. We will demonstrate the
advantage of the LDA surpassing PCA under a small number of qubits and with
relevant features.

Figure 1: High-level workflow of the process for the hybrid quantum approach
followed in this research. The input is a dataset in CSV format that is
analyzed and divided depending on the following decomposition strategies to be
applied. The next step is dimensionality reduction using different techniques
but mainly PCA and LDA as the main comparison in this exploration. Once the
reduction is executed - to match with the qubits to be used in the algorithm -
a quantum encoding must be conducted and the following quantum algorithm
applied to extract the results.]

###  3.2 Training

The interest in QML is located in the quantum advantage that can be brought by
such new technology. A quantum advantage appears when the quantum algorithm
approach provides better or faster results than the classical equivalent. In
this paper we benchmark classical classifiers such as Logistic RegressionHaug
et al. (2021), Decision TreeSen (2021), Random ForestSen (2021), K-Nearest
NeighborsSen (2021), SVMSen (2021), Quantum Kernel (QSVC)Torabian and Krems
(2022), and Variational Quantum AlgorithmChen et al. (2020). The models are
benchmarked with k-fold cross-validation with 10 folds. Table 1 shows the
metrics used to evaluate these algorithms.

Metric | Equation
---|---
Precision | $\frac{TP}{TP+FP}$
Recall | $\frac{TP}{TP+FN}$
F1-score | $\frac{2\times(Precision\times Recall)}{Precision+Recall}$
MCC | $\frac{TP\times TN\textendash FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}$
BA | $\left(\frac{TP}{(TP+FN)}/\frac{TN}{(TN+FP)}\right)/2$
Table 1: Metrics are used to evaluate the different classifiers and their
corresponding equation. Where TP, FP, TN, and FN are True Positive, False
Positive, True Negative, and False Positive, respectively. $P_{o}$ is the
probability of Observed agreement, and $P_{c}$ is the probability of chance
agreement. MCC is the Matthews correlation coefficient.

##  4 Backends

Quantum computers and simulator backends are not trivial decisions when a fast
iteration is needed with larger datasets. Typically machine learning models
need several adjustments and iterations until we put them in production or
under real-world operation (fine tuning). In the case of QML, the challenge is
the same, but the hardware ecosystem is different. A quantum algorithm can be
run on a simulator (simulation of a perfect and noisy quantum computer) and
real devices.

###  4.1 Simulators

Mainly the use of quantum simulators allows us to test and evaluate the
results under potential real quantum computation scenarios and typically gives
us the chance to operate up to 50 qubits using classical computers. In the
case of this experiment, we are using the Qiskit Aer simulator and the default
qubit simulator device from Pennylane (Bergholm et al., 2018) only.

Qiskit Aer is a high-performance simulator for Qiskit Terra that provides a
highly adjustable noisy model for investigating quantum computing in the NISQ
domain. The core is designed in C++ for speed and includes elements from IBM’s
high-performance online simulators into a local simulator that is scaled to
operate even on your own laptop or server.

In the case of Pennylane’s default qubit, is a simple state-vector qubit
simulator designed in Python with JAX, Autograd, Torch, and Tensorflow. This
simulator is recommended by Pennylane for optimizations with a reduced number
of qubits or when stochastic expectation values are going to be used.

The objective behind using simulators alone in this research is that we can
have fast iterations, adjustments, and results. Our choice does not mean that
our models’ code and structure cannot be applied to real quantum hardware. The
main factors that make the difference are the time (speed), and the noise
since simulators run in a fraction of the time compared with real quantum
computers (in the case of the QML architectures that were used in this
investigation) and also without noise depending on the configuration. In our
case, we subtracted the noise from the simulations.

##  5 Algorithms

###  5.1 Machine Learning Models

Classification problems are part of the supervised learning domain and that’s
why we used several classical algorithms in this subarea of ML to set a
benchmark against the hybrid quantum-classical approach.

####  5.1.1 Logistic Regression

This method is one of the simplest for the binary classification problem
(Cramer, 2002). The model is train to learn the parameters of the linear
equation
$\hat{y}^{(i)}=\beta_{0}+\beta_{1}x_{1}^{(i)}+\beta_{2}x_{2}^{(i)}+...+\beta_{n}x_{n}^{(i)}$
where $\beta_{n}$ are the coefficients of the linear regression, $x_{n}$ are
the features for the $i$ sample. Linear regression fails in the classification
task is why the linear regression formulation is embedded in a logistic
function such as Eq. 1 to compute a probability.

$P(y^{(i)}=1)=\frac{1}{1+e^{-(\beta_{0}+\beta_{1}x_{1}^{(i)}+...+\beta_{n}x_{n}^{(i)})}}$
(1)

$P$ are the probability that the label $y$ for the sample $i$ corresponds to
the value 1. The probability is computed for each sample (the model learns the
corresponding coefficients) and the probability threshold is fixed at 0.5 to
separate the binary outcome. If $P(y^{(i)}=1)<0.5$ the corresponding label is
0, if $P(y^{(i)}=1)\geq 0.5$ the label is 1. The logistic regression method
requires a lot of samples to be stable to efficiently approximate the
$\beta_{n}$ parameters.

####  5.1.2 Decision Tree

A decision tree (CART) is a sort of binary graph where the next child is based
on the previous decision. The base of the tree is the root, and then two
branches are created called split (as "yes" or "no"). A tree structure is
built by successive decisions until the latest called the leaf is reached.
This type of technique is simple but prone to overfitting. They are powerful
algorithms capable of fitting complex datasets. The learning process is done
with the Gini criteria or entropy (Eq. 2).

$H_{i}=\sum_{k=1}^{n}P_{i,k}\log_{2}{P_{i,k}}$ (2)

where $i$ is the $i^{th}$ node, $P_{i,k}$ is the probability of the category
$k$.

####  5.1.3 Naïve Bayes

The Naïve Bayes or NB algorithm is a simpler version of the Bayes theorem (Eq.
3).

$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$ (3)

Where $A$ and $B$ are events, $P(A|B)$ is the probability of $A$ given $B$ is
true, $P(B|A)$ is the probability of $B$ given $A$ is true, $P(A)$ and $P(B)$
are the independent probabilities of A and B respectively. In the case of the
NB classifier, the probabilities are conditionally independent. It
significantly reduced the computation and transformed it into a tractable
problem.

####  5.1.4 k-Nearest Neighbors

The k-Nearest Neighbors (k-NN) is a simple non-parametric distance-based
algorithm. The hypothesis is that a similar point will be closed in an
n-dimensional space. A point will be encoded and positioned by distance
computation (e.g. Euclidean distance). Then, the algorithm takes the $k$
nearest neighbours and computes the classes’ average to predict the
corresponding class for that new point.

####  5.1.5 SVM

Support Vector Machines (SVM) are a class of algorithms based on class
separation by a plan. An SVM will create a plan to create a binary separation
between the classes. Then, the algorithm will compute the distance of each
point and plan to maximize the distance. When the classes are not linearly
separable, SVM can be used with kernels. Kernels are a trick to compute small
successive plans to separate classes in complex datasets. They are
particularly efficient in high-dimensional space.

###  5.2 Dimensionality reduction

####  5.2.1 SVD

Singular Value Decomposition or SVD was established to decompose the matrix
representation of data into distinct matrices. It’s a factorization process
for real and complex matrices. These transformations are based on eigenvalue
decomposition to diagonalize a matrix.

####  5.2.2 PCA

Principal component analysis or PCA is a widely used method for dimensionality
reduction in the context of machine learning. The objective is to transform a
large dataset (a high number of features) into a compact representation
containing the data’s important information (orthogonal projection). Reducing
the dimension is closely related to loss in accuracy, but PCA as SVD uses the
eigenvalue decomposition process to transform the covariance matrix involved
in the process. The components are a linear combination of various features to
create uncorrelated new features. The first component will contain the maximum
of the information, then the remaining in the second etc. The geometrical
representation of PCA is that components represent the direction of the
maximal amount of variance (rotation).

####  5.2.3 SKPP

Projection pursuit is a generalization of PCABarcaru (2019) where the method
aims to find the best projections through the feature to maximize or minimize
a projection index. To find the relevant project index, the method uses the
Kurtosis-based projectionHou and Wentzell (2011). In a case of a supervised
dataset, the algorithm is named Supervised Kurtosis Projection Pursuit (SKPP).
The Kurtosis index can be expressed as:

$K=\frac{1/n\sum_{i=1}^{n}(z_{i}-\tilde{z})^{4}}{\left(1/n\sum_{i=1}^{n}(z_{i}-\tilde{z})^{2}\right)^{2}}$
(4)

where $n$ is the number of samples, $z_{i}$ is the individual sample value,
and $\tilde{z}$ is the sample mean. The numerator is the fourth central
moment, and the denominator is the biased sample variance.

####  5.2.4 LDA

Linear Discriminant Analysis is similar to PCA, they are linear
transformations to reduce the dimensionality of datasets (eigenvalue
decomposition). Where PCA will maximize the variance, LDA will maximize the
axes for class separation. LDA will create a subspace of k-dimensions from the
n-dimensions space of the original data where $k\leq n-1$. The subspace is
computed taking into account the label to maximize the separation of classes.

###  5.3 Quantum Machine Learning Models

Quantum Computing (QC) and Machine Learning (ML) have been mixed to develop a
new area which is Quantum Machine Learning (QML). This new field of study
incorporates ideas from both aspects to provide better answers by boosting the
performance of either ML algorithms or quantum experiments, or both. By
utilizing quantum resources to increase machine learning in terms of speed
and/or performance, researchers could obtain potential alternative and/or more
accurate solutions.

####  5.3.1 Quantum Kernel

The quantum kernel methods Schuld (2021); Guo and Weng (2022) in principle are
the same as their classical versions which aim to classify data by defining
what is similar in a given space because of their distance using a feature
mapping function. The main difference in the quantum version of kernels is
that it maps out the data points from the original input to a high-dimensional
Hilbert feature space expanding the possibilities to find the best
classification possible (Havlíček et al., 2019). One of the well-known
classical methods that utilize the kernel properties is the support vector
machine (SVM) or also known as support vector classifier (SVC) which is
dedicated to finding a hyper-plane that can separate the classes of the
datapoints expanding as much as is possible the distance between both groups.

In this research, we use a similar structure of an SVC but boosted with a
quantum kernel. The mechanism of quantum kernel functions resembles the
conventional one, but its implementation relies on quantum superposition
states and entanglement. Also in the case of quantum kernels, the output
values are statistically dependent on probabilities, so some researchers call
this method a probabilistic kernel function.

####  5.3.2 Variational Quantum Classifier

A Variational Quantum Classifier (VQC) (Pérez-Salinas et al., 2020; Yano et
al., 2021; Li and Deng, 2022) is a supervised quantum-classical hybrid method
widely used in NISQ devices and simulators. The cost function in the case of
this algorithm is calculated using iterative measurements, which also provide
the possibility of mitigating errors. This method allows the researcher and
developers to map classical data and grab benefits for an increasingly ample
feature space in quantum. The quantum execution for supervised learning
employs variational algorithms that are implemented using differential
programming, state preparation that encodes classical data sets into amplitude
and rotations of qubits for quantum hardware or simulators to comprehend, and
qubits that are executed using parameterized unitary operations; all
parameters are modifiable according to given rules. The outcome of the quantum
execution is the output, which categorizes the input data.

###  5.4 Quantum encoding

Quantum encoding is the process of passing from classical data to quantum
representations. There are many ways to process classical data and create a
useful representation. In this study, we used a quantum feature map (Qiskit
ZZFeatureMap) and angle encoding (Pennylane) to be used with QSVC and VQC
respectively.

####  5.4.1 Quantum feature map

A feature map is a new representation for data encoding (Schuld and Killoran,
2019; Havlíček et al., 2019; Altares-López et al., 2021; Huang et al., 2021))
and it is represented as a Hilbert space where the transformation
$x\rightarrow|\phi(x)\rangle$ is applied to pass from the original
representation $x$ into a linear separable space via a unitary operator
$\phi(x)$. In other words, we encode the classical data into quantum states
and map them to Hilbert space.

The feature map is a good (sometimes infinite) projection for using SVM that
is designed to create a hyperplane between classes. This hyperplane is a
linear separation of two subspaces that are automatically created by the
feature map (different data projections make the separation easier with higher
dimensions).

####  5.4.2 Angle encoding

Angle encoding is a process of encoding classical information by rotations
(Schuld and Petruccione, 2018). The classical information is represented by
angles of rotation in corresponding gates and can be written as Eq. 5:

$|x\rangle=\bigotimes_{i}^{n}R(x_{i})|0^{n}\rangle$ (5)

Where $R$ is rotation gates such as $R_{x}$, $R_{y}$ and $R_{z}$. Angle
encoding is used when the dimension of the feature vector $x$ is equal to the
number of qubits.

###  5.5 Workflow

In Figure 1, we present the workflow we used throughout this study to compare
the selected algorithms (classical and quantum). The set of algorithms was
applied to the data representation generated by dimensionality reduction
methods. The workflow is composed of five steps:

  * 1-2

Steps 1 and 2 can be associated: Load the data and apply an Exploratory Data
Analysis. The objective is to clean the data and normalize it with a good
format for the dimensionality reduction method.

  * 3

Dimensionality reduction: SVD, PCA, SKPP and LDA are used to reduce the number
of features to two compressed dimensions. SVD, PCA and SKPP were used with two
components. LDA was used on a split dataset. Each half part was reduced with
one component by LDA.

  * 4

Quantum encoding: The classical data is encoded into a quantum representation
by quantum feature maps. This step was only used for quantum algorithms.

  * 5

Applied models: The selected set of algorithms (ML and QML) is applied to the
data encoding (classical or quantum) and evaluated through the same metrics
(Table 1)

During the workflow, we evaluate the set of selected algorithms based on the
same sample of data. 800 data samples were used for the training process, and
200 were used for the test. Only two qubits were chosen through this study to
show the usability of a small number of qubits in a business context. Two
datasets close to the real world were selected to estimate the importance of
current quantum algorithms with NISQ devices.

##  6 Results

This section will present the results obtained by applying the workflow to the
two datasets we selected. The core of the analysis is to take the position of
the business context. The classical machine learning algorithms are applied to
both datasets without dimensionality reduction and serve as a baseline. Then,
the quantum algorithms are applied to a subset of each dataset with diverse
dimensionality reduction techniques. This choice is motivated by focusing on
only two qubits to compare the results. Indeed, current commercial solutions
provide quantum computers with two qubits. Also, cloud-free available quantum
computers are up to five qubits.

We focus on a small number of qubits to demonstrate the usability of quantum
algorithms in a business context. The quantum version of SVM (QSVC) and a
variational quantum algorithm (VQA) have been used to challenge the classical
machine learning methods. The subsample is constituted of 800 samples for the
training phase and 200 for the test phase. Each model, classical and quantum
will be evaluated with the metrics presented in Table 1. Each metric for each
algorithm is associated with an error bar determined by a k-fold cross-
validation approach with 10 folds. Only the VQA was computed differently due
to its implementation, and it is not provided with an associative error.

We will analyze the results for both datasets separately in the following
subsections. Classical machine learning models were also applied to the same
sample with the same dimensionality reduction approaches, the results are
provided for comparison in Appendix A. These results are discussed in section
7 Discussion.

###  6.1 UCI Credit Card default dataset

Table 2 shows the results for the baseline determined with the classical
machine learning models. LR and SVM show a non-convergence state with
predictions only for the majority class. Naïve Bayes classifier shows the
smallest precision compare to CART and KNN. The baseline demonstrates that
classical methods struggle to create a good classifier to separate the
minority class from the majority class. The precision is at a maximum of
38.74% for the KNN, and the best f1-score is reached by CART with a small
value of 39.10%. These poor metrics are the results of an extremely imbalanced
dataset. Little information is provided by the minority class, which tends to
complexify the classification process. Machine Learning methods are best
suited for a balanced dataset, but this condition is rarely present in the
industry.

Table 3 shows the results for QSVC and VQA applied to the UCI Credit card
dataset. They are used with SVD, PCA, SKPP and LDA dimensionality reduction
techniques. QSVC with the SKPP technique shows metrics with 0.00%, meaning a
non-separation between both classes. QSVC, in this case, predicts only the
majority class output, ignoring the minority class. The other algorithms
associated with the different dimensionality reduction approaches show a
convergence. VQA with SVD, VQA with PCA, and VQA with SKPP provide interesting
results comparable to or better than the baseline. LDA is the best
dimensionality reduction for both quantum algorithms. Both algorithms show the
best results in each metric on the UCI dataset.

Figure 3 shows a histogram representation of the metrics for both the ML and
QML approaches. Figure 2 shows the difference between LDA and PCA methods for
the UCI Credit Card dataset. The LDA representation provides a net advantage
for quantum encoding. The use of LDA shows a quantum advantage for QSVC and
VQA algorithms.

| Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | -0.22 (0.44) | 49.99 (0.01)
KNN |  38.74 (2.03) | 15.45 (1.51) | 22.07 (1.76) | 12.43 (0.76) | 54.26 (0.65)
CART | 37.79 (1.51) | 40.53 (1.51) |  39.10 (1.34) |  20.99 (1.45) |  60.76 (0.75)
NB | 24.71 (0.89) |  88.41 (1.55) | 38.62 (1.15) | 11.94 (1.74) | 55.82 (0.88)
SVM | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 50.00 (0.00)
Table 2: Baseline ML models for the UCI Credit Card default dataset. The results for logistic regression (LR), k-Nearest Neighbors (KNN), Naïve Bayes (NB) and Support Vector Machine (SVM) are computed with k-fold cross-validation with 10 folds.  | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
QSVC (SVD) | 20.00 (40.00) | 2.21 (4.82) | 3.92 (8.45) | 5.98 (12.30) | 51.10 (2.41)
VQA (SVD) | 77.50 | 26.72 | 39.74 | 19.75 | 58.00
QSVC (PCA) | 12.00 (29.93) | 1.06 (2.14) | 1.88 (3.84) | 0.51 (8.04) | 49.93 (1.30)
VQA (PCA) | 88.10 | 25.87 | 40.00 | 18.95 | 58.55
QSVC (SKPP) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
VQA (SKPP) | 25.58 | 27.5 | 26.51 | 7.3 | 53.75
QSVC (LDA) | 67.02 (13.31) | 33.44 (10.08) | 43.96 (10.97) | 38.51 (10.97) | 64.6 (5.08)
VQA (LDA) | 41.30 | 100.00 | 58.46 | 59.28 | 92.54
Table 3: Quantum models applied on the UCI Credit Card default dataset with
the corresponding dimensionality methods. LDA PCA
VQCQSVCVQCQSVC$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Precision
Recall f1 ScoreMatthews corcorrefBalanced Accuracy Figure 2: Metrics
comparison between VQC and QSVC using LDA and PCA applied on UCI Credit Card
default dataset. Quantum Classical
VQCQSVCCARTKNN$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Precision
Recall f1 ScoreMatthews corcorrefBalanced Accuracy Figure 3: Metrics
comparison of VQC and QSVC with CART and KNN (best classical algorithms) with
the application of LDA using UCI credit cards dataset.

###  6.2 Fraud (bank) Detection Dataset

Table 4 shows the results for the baseline for the fraud (bank) detection
dataset with classical ML. The precision is up to 70% for LR, KNN, and CART
but NB has a precision of less than 30% and SVM shows metrics with 0.00%.
Table 5 show the results for the quantum algorithms with the different
dimensionality reduction techniques. PCA demonstrates the worst representation
for QSVC (Figure 4). The algorithms didn’t converge. LDA shows the best
results for both QSVC and VQA. SKPP also demonstrates interesting results for
both algorithms. It’s worth noting that four quantum algorithms beat the best
precision value of the baseline and the other metrics are close to the
classical ML models.

Even if the baseline classical ML models perform well, the quantum algorithms
provide a small advantage over the LDA approach (as the UCI Credit Card fraud
dataset results). Figure 5 shows VQA and QSVC compared to CART and KNN
(metrics representation).

| Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 71.54 (2.77) | 47.27 (1.96) | 56.88 (1.62) | 46.89 (1.88) | 70.2 (0.88)
KNN | 74.34 (1.77) | 64.56 (2.36) | 69.09 (1.91) | 59.16 (2.65) | 78.22 (1.39)
CART |  80.68 (1.87) | 81.69 (2.06) |  81.17 (1.63) |  74.27 (2.25) |  87.28 (1.21)
NB | 28.43 (1.07) |  96.95 (0.88) | 43.96 (1.3) | 12.58 (1.36) | 54.07 (0.54)
SVM | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
Table 4: Baseline ML models for the fraud (bank) detection dataset. The results for logistic regression (LR), k-Nearest Neighbors (KNN), Naïve Bayes (NB) and Support Vector Machine (SVM) are computed with k-fold cross-validation for 10 folds.  | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
QSVC (SVD) | 85.02 (11.42) | 39.24 (8.53) | 52.94 (8.19) | 49.55 (7.54) | 68.45 (3.97)
VQA (SVD) | 62.50 | 72.22 | 60.61 | 26.57 | 74.09
QSVC (PCA) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
VQA (PCA) | 67.39 | 25.41 | 36.90 | 7.16 | 53.09
QSVC (SKPP) | 56.28 (11.17) | 46.46 (7.21) | 50.3 (6.8) | 35.53 (8.7) | 66.65 (4.02)
VQA (SKPP) | 89.86 | 68.89 | 77.99 | 70.67 | 82.60
QSVC (LDA) | 82.35 (10.29) | 65.92 (8.79) | 72.93 (8.14) | 66.35 (9.9) | 80.67 (4.94)
VQA (LDA) | 84.00 | 84.44 | 75.68 | 55.81 | 83.92
Table 5: Quantum models applied on the fraud (bank) detection dataset in
combination with the corresponding dimensionality methods.  LDA PCA
VQCQSVCVQCQSVC$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Precision
Recall f1 ScoreMatthews corcorrefBalanced Accuracy Figure 4: Metrics
comparison between VQC and QSVC using LDA and PCA applied on fraud (bank)
detection dataset. Quantum Classical
VQCQSVCCARTKNN$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Precision
Recall f1 ScoreMatthews corcorrefBalanced Accuracy Figure 5: Metrics
comparison of VQC and QSVC with CART and KNN (best classical algorithms) with
the application of LDA using fraud (bank) detection dataset.

##  7 Discussion and conclusion

In this study, we demonstrate that specific techniques of preprocessing could
play a crucial role when we talk about quantum machine learning. We focus on
the encoding part, the classical one, to evaluate the effect on quantum
algorithms. We show that a quantum computer can extract more meaningful
information from classical data and leverage classification results just using
a few dimensions. As postulated in Schuld and Killoran (2022) quantum
advantage doesn’t need to be measured by the ability to beat classical ML
models but can be regarded as a better information extraction technique. The
few numbers of qubits of the currently accessible quantum computers force
researchers to look to new alternatives. Classical dimensionality reduction as
SKPP, PCA or LDA are useful to compress classical high feature datasets (100+)
into a number that can be used with a quantum computer. Here, we tested with
two dimensions for two qubits. LDA shows more promising results for supervised
machine learning tasks with quantum computers. The prevalence of LDA under PCA
wasn’t explored in this paper but will be explored in future to understand how
LDA provides a better data representation for qubits encoding. Further
analysis will be needed to determine the positive effect of LDA in supervised
QML. Also, we will study the potential impact of PCA in unsupervised tasks. As
in classical ML, we need to determine which methods has a better effect on
specific types of data.

Tables 6, 7, 8, 9, 10, 11, 12, 13 show the results where the classical methods
were also applied with the dimensionality reduction methods. The DR also
improves the performances of these methods, but quantum algorithms are
comparable to them. More investigation will be needed, but the preprocessing
part of big real-world datasets plays an important role in the usability of
quantum computing in the industry. Better quantum data encoding will also be
required to demonstrate a strong difference between classical and quantum
machine learning.

Classical ML methods selected for this study weren’t tuned specifically on the
two datasets. Default parameters were chosen. Only LR was used with a max
iteration parameter fixed at 1,000 iterations. Indeed, on the fraud detection
dataset, the default limit was reached and LR wasn’t converging. KNN was
trained with the number of neighbours fixed at seven. In the case of quantum
algorithms, the data is encoded with a feature map. The parameters of the
feature map weren’t tuned but fixed through the study with a number of
repetitions of 2 and a feature dimension of 2. Further investigation is needed
to explore the effect of the feature map on the output of the QSVC. VQA was
used with an angle embedding method (rotations gates) and a strongly entangled
layer. Both weren’t tuned, and alternatives will be explored in the future. An
interesting perspective will also to study the impact of different ansatzes
(Cerezo et al., 2021; Ostaszewski et al., 2021) on the VQA results.

Also, in this study, we used only two datasets close to the real-world data in
the finance domain. The results demonstrate a quantum advantage with QSVC and
VQA, but we need to extend the approach with other datasets (higher number of
features) in other domains to create a benchmark through a general
application. We demonstrate that the quantum era needs to be seriously
investigated by the industrial people, but more work is needed to fully
demonstrate the advantages.

## Code availability

The notebooks used during this study can be provided after contacting the
authors.

## References

  * Provenzano et al. [2020] A. R. Provenzano, D. Trifirò, A. Datteo, L. Giada, N. Jean, A. Riciputi, G. Le Pera, M. Spadaccino, L. Massaron, and C. Nordio.  Machine learning approach for credit scoring.  2020\.  doi: 10.48550/ARXIV.2008.01687.  URL https://arxiv.org/abs/2008.01687.
  * Tiwari et al. [2021] Pooja Tiwari, Simran Mehta, Nishtha Sakhuja, Jitendra Kumar, and Ashutosh Kumar Singh.  Credit card fraud detection using machine learning: A study.  _CoRR_ , abs/2108.10005, 2021.  URL https://arxiv.org/abs/2108.10005.
  * Rohde et al. [2018] David Rohde, Stephen Bonner, Travis Dunlop, Flavian Vasile, and Alexandros Karatzoglou.  Recogym: A reinforcement learning environment for the problem of product recommendation in online advertising.  _CoRR_ , abs/1808.00720, 2018.  URL http://arxiv.org/abs/1808.00720.
  * Masini et al. [2020] Ricardo P. Masini, Marcelo C. Medeiros, and Eduardo F. Mendes.  Machine learning advances for time series forecasting.  2020\.  doi: 10.48550/ARXIV.2012.12802.  URL https://arxiv.org/abs/2012.12802.
  * Mishra et al. [2021] Nimish Mishra, Manik Kapil, Hemant Rakesh, Amit Anand, Nilima Mishra, Aakash Warke, Soumya Sarkar, Sanchayan Dutta, Sabhyata Gupta, Aditya Dash, Rakshit Gharat, Yagnik Chatterjee, Shuvarati Roy, Shivam Raj, Valay Jain, Shreeram Bagaria, Smit Chaudhary, Vishwanath Singh, Rituparna Maji, and Prasanta Panigrahi.  _Quantum Machine Learning: A Review and Current Status_ , pages 101–145.  01 2021.  ISBN 978-981-15-5618-0.  doi: 10.1007/978-981-15-5619-7_8.
  * Ristè et al. [2013] Diego Ristè, CC Bultink, Marijn J Tiggelman, Raymond N Schouten, Konrad W Lehnert, and Leonardo DiCarlo.  Millisecond charge-parity fluctuations and induced decoherence in a superconducting transmon qubit.  _Nature communications_ , 4(1):1–6, 2013.
  * Burnett et al. [2019] Jonathan J Burnett, Andreas Bengtsson, Marco Scigliuzzo, David Niepce, Marina Kudra, Per Delsing, and Jonas Bylander.  Decoherence benchmarking of superconducting qubits.  _npj Quantum Information_ , 5(1):1–8, 2019.
  * Wang et al. [2021] Samson Wang, Enrico Fontana, Marco Cerezo, Kunal Sharma, Akira Sone, Lukasz Cincio, and Patrick J Coles.  Noise-induced barren plateaus in variational quantum algorithms.  _Nature communications_ , 12(1):1–11, 2021.
  * Preskill [2018] John Preskill.  Quantum Computing in the NISQ era and beyond.  _Quantum_ , 2:79, August 2018.  ISSN 2521-327X.  doi: 10.22331/q-2018-08-06-79.  URL https://doi.org/10.22331/q-2018-08-06-79.
  * Callison and Chancellor [2022] Adam Callison and Nicholas Chancellor.  Hybrid quantum-classical algorithms in the noisy intermediate-scale quantum era and beyond.  _Physical Review A_ , 106(1), jul 2022.  doi: 10.1103/physreva.106.010101.  URL https://doi.org/10.1103%2Fphysreva.106.010101.
  * CaixaBank [2022] CaixaBank.  For access to our content, please go to https://www.euromoney.com/reprints.  _Euromoney_ , 2022.  URL https://www.euromoney.com/article/2a8fb2jukdh3ae709zojl/sponsored-content/quantum-banking-the-next-leap-in-financial-services.
  * Chow et al. [2021] Jerry Chow, Oliver Dial, and Jay Gambetta.  Ibm quantum breaks the 100-qubit processor barrier.  _IBM Blog_ , 2021.  URL https://research.ibm.com/blog/127-qubit-quantum-processor-eagle.
  * Dalzell et al. [2020] Alexander M. Dalzell, Aram W. Harrow, Dax Enshan Koh, and Rolando L. La Placa.  How many qubits are needed for quantum computational supremacy?  _Quantum_ , 4:264, may 2020.  doi: 10.22331/q-2020-05-11-264.  URL https://doi.org/10.22331%2Fq-2020-05-11-264.
  * Haug et al. [2021] Tobias Haug, Chris N. Self, and M. S. Kim.  Large-scale quantum machine learning, 2021.  URL https://arxiv.org/abs/2108.01039.
  * Yeh and Lien [2009] I-Cheng Yeh and Che-hui Lien.  The comparisons of data mining techniques for the predictive accuracy of probability of default of credit card clients.  _Expert systems with applications_ , 36(2):2473–2480, 2009.
  * Mensa et al. [2022] Stefano Mensa, Emre Sahin, Francesco Tacchino, Panagiotis Kl. Barkoutsos, and Ivano Tavernelli.  Quantum machine learning framework for virtual screening in drug discovery: a prospective quantum advantage, 2022.  URL https://arxiv.org/abs/2204.04017.
  * Sen [2021] Jaydip Sen, editor.  _Machine Learning - Algorithms, Models and Applications_.  IntechOpen, dec 2021.  doi: 10.5772/intechopen.94615.  URL https://doi.org/10.5772%2Fintechopen.94615.
  * Torabian and Krems [2022] Elham Torabian and Roman V. Krems.  Optimal quantum kernels for small data classification, 2022.  URL https://arxiv.org/abs/2203.13848.
  * Chen et al. [2020] Samuel Yen-Chi Chen, Chih-Min Huang, Chia-Wei Hsing, and Ying-Jer Kao.  Hybrid quantum-classical classifier based on tensor network and variational quantum circuit, 2020.  URL https://arxiv.org/abs/2011.14651.
  * Bergholm et al. [2018] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, M Sohaib Alam, Shahnawaz Ahmed, Juan Miguel Arrazola, Carsten Blank, Alain Delgado, Soran Jahangiri, et al.  Pennylane: Automatic differentiation of hybrid quantum-classical computations.  _arXiv preprint arXiv:1811.04968_ , 2018.
  * Cramer [2002] J.S. Cramer.  The Origins of Logistic Regression.  Tinbergen Institute Discussion Papers 02-119/4, Tinbergen Institute, December 2002.  URL https://ideas.repec.org/p/tin/wpaper/20020119.html.
  * Barcaru [2019] Andrei Barcaru.  Supervised projection pursuit – a dimensionality reduction technique optimized for probabilistic classification.  _Chemometrics and Intelligent Laboratory Systems_ , 194:103867, 2019.  ISSN 0169-7439.  doi: https://doi.org/10.1016/j.chemolab.2019.103867.  URL https://www.sciencedirect.com/science/article/pii/S0169743919300309.
  * Hou and Wentzell [2011] S. Hou and P.D. Wentzell.  Fast and simple methods for the optimization of kurtosis used as a projection pursuit index.  _Analytica Chimica Acta_ , 704(1):1–15, 2011\.  ISSN 0003-2670.  doi: https://doi.org/10.1016/j.aca.2011.08.006.  URL https://www.sciencedirect.com/science/article/pii/S0003267011010804.
  * Schuld [2021] Maria Schuld.  Supervised quantum machine learning models are kernel methods.  _arXiv preprint arXiv:2101.11020_ , 2021.
  * Guo and Weng [2022] Muhao Guo and Yang Weng.  Where can quantum kernel methods make a big difference?, 2022.  URL https://openreview.net/forum?id=NoE4RfaOOa.
  * Havlíček et al. [2019] Vojtěch Havlíček, Antonio D Córcoles, Kristan Temme, Aram W Harrow, Abhinav Kandala, Jerry M Chow, and Jay M Gambetta.  Supervised learning with quantum-enhanced feature spaces.  _Nature_ , 567(7747):209–212, 2019.
  * Pérez-Salinas et al. [2020] Adrián Pérez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and José I Latorre.  Data re-uploading for a universal quantum classifier.  _Quantum_ , 4:226, 2020.
  * Yano et al. [2021] Hiroshi Yano, Yudai Suzuki, Kohei Itoh, Rudy Raymond, and Naoki Yamamoto.  Efficient discrete feature encoding for variational quantum classifier.  _IEEE Transactions on Quantum Engineering_ , 2:1–14, 2021\.  doi: 10.1109/tqe.2021.3103050.  URL https://doi.org/10.1109%2Ftqe.2021.3103050.
  * Li and Deng [2022] Weikang Li and Dong-Ling Deng.  Recent advances for quantum classifiers.  _Science China Physics, Mechanics & Astronomy_, 65(2):1–23, 2022.
  * Schuld and Killoran [2019] Maria Schuld and Nathan Killoran.  Quantum machine learning in feature hilbert spaces.  _Physical review letters_ , 122(4):040504, 2019\.
  * Altares-López et al. [2021] Sergio Altares-López, Angela Ribeiro, and Juan José García-Ripoll.  Automatic design of quantum feature maps.  _Quantum Science and Technology_ , 6(4):045015, aug 2021.  doi: 10.1088/2058-9565/ac1ab1.  URL https://doi.org/10.1088/2058-9565/ac1ab1.
  * Huang et al. [2021] Hsin-Yuan Huang, Michael Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, and Jarrod R McClean.  Power of data in quantum machine learning.  _Nature communications_ , 12(1):1–9, 2021.
  * Schuld and Petruccione [2018] Maria Schuld and Francesco Petruccione.  _Supervised Learning with Quantum Computers_.  Springer Publishing Company, Incorporated, 1st edition, 2018.  ISBN 3319964232.
  * Schuld and Killoran [2022] Maria Schuld and Nathan Killoran.  Is quantum advantage the right goal for quantum machine learning?  _PRX Quantum_ , 3:030101, Jul 2022.  doi: 10.1103/PRXQuantum.3.030101.  URL https://link.aps.org/doi/10.1103/PRXQuantum.3.030101.
  * Cerezo et al. [2021] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al.  Variational quantum algorithms.  _Nature Reviews Physics_ , 3(9):625–644, 2021\.
  * Ostaszewski et al. [2021] Mateusz Ostaszewski, Edward Grant, and Marcello Benedetti.  Structure optimization for parameterized quantum circuits.  _Quantum_ , 5:391, 2021.

##  Appendix A All results

We applied the classical ML algorithms with the reduced data representation
after the different dimensionality reduction approaches. The next sections
will provide a table per dimensionality reduction technique for both datasets
with all the algorithms applied.

###  A.1 UCI Credit Card fraud dataset

The Tables 6, 7, 8, 9 provide the performance by metrics for classical machine
learning and quantum algorithms applied on the same sample with dimensionality
reduction techniques.

| Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 30.00 (45.83) | 1.70 (2.61) | 3.22 (4.93) | 3.53 (11.92) | 50.50 (1.57)
KNN | 55.49 (8.66) | 36.86 (10.08) |  43.06 (7.05) | 31.20 (6.48) |  63.47 (3.73)
CART | 36.35 (9.50) |  38.49 (13.49) | 36.84 (11.02) | 15.86 (11.91) | 57.70 (5.23)
NB | 73.27 (19.89) | 17.36 (10.28) | 25.78 (11.65) | 25.23 (8.60) | 57.07 (4.12)
SVM | 69.94 (15.02) | 26.29 (9.15) | 37.24 (10.20) |  33.27 (9.59) | 61.34 (4.12)
QSVC | 20.00 (40.00) | 2.21 (4.82) | 3.92 (8.45) | 5.98 (12.30) | 51.10 (2.41)
VQA | 77.5 | 26.72 | 39.74 | 19.75 | 58.00
Table 6: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA) applied to the UCI Credit Card fraud dataset using the SVD dimensionality reduction. | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 50.00 (0.00)
k-NN | 35.70 (7.98) | 20.98 (7.05) | 26.07 (7.78) | 10.72 (9.61) | 54.49 (3.98)
CART | 30.02 (7.31) |  34.82 (10.15) | 33.55 (8.89) | 9.69 (9.87) | 54.87 (4.31)
NB | 18.33 (18.93) | 8.63 (12.14) | 10.22 (12.45) | 4.01 (6.56) | 51.61 (2.38)
SVC | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 0.00 (0.00) | 50.00 (0.00)
QSVC | 12.00 (29.93) | 1.06 (2.14) | 1.88 (3.84) | 0.51 (8.04) | 49.93 (1.30)
VQA | 88.10 | 25.87 | 40.00 | 18.95 | 58.55
Table 7: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA) applied to the UCI Credit Card fraud dataset using the PCA dimensionality reduction. | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 74.12 (15.51) | 28.58 (8.58) | 40.66 (10.14) | 37.16 (11.54) | 62.64 (4.72)
KNN | 59.37 (9.33) | 36.18 (9.87) |  43.85 (8.48) | 33.71 (8.71) |  63.97 (4.67)
CART | 34.4 (8.39) |  36.7 (13.78) | 35.13 (10.53) | 15.15 (13.59) | 57.94 (7.37)
NB | 66.66 (14.72) | 25.42 (9.45) | 36.27 (11.71) | 31.74 (12.55) | 60.82 (5.00)
SVM |  83.45 (16.05) | 28.62 (8.78) | 41.83 (10.09) |  41.00 (11.03) | 63.23 (4.59)
QSVC | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
QVA | 25.58 | 27.5 | 26.51 | 7.3 | 53.75
Table 8: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA) applied to the UCI Credit Card fraud dataset using the SKPP dimensionality reduction. | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 64.75 (37.02) | 12.36 (9.46) | 19.89 (14.14) | 22.66 (14.89) | 55.61 (4.39)
KNN | 57.06 (20.44) | 33.36 (12.55) | 40.71 (12.24) | 32.17 (14.4) | 62.99 (6.6)
CART | 34.83 (9.49) | 34.12 (11.25) | 33.93 (9.39) | 17.25 (10.27) | 58.6 (5.34)
NB | 62.58 (15.57) | 39.13 (9.38) | 47.41 (10.23) | 39.19 (11.71) | 66.25 (5.07)
SVM | 66.64 (11.82) | 22.86 (9.57) | 33.2 (11.84) | 30.9 (10.38) | 60.01 (4.51)
QSVC |  67.02 (13.31) | 33.44 (10.08) | 43.96 (10.97) | 38.51 (10.97) | 64.6 (5.08)
VQA | 41.30 | 100.00 | 58.46 | 59.28 | 92.54
Table 9: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA)
applied to the UCI Credit Card fraud dataset using the LDA dimensionality
reduction.

###  A.2 Fraud (bank) detection dataset

The Tables 10, 11, 12, 13 provide the performance by metrics for classical
machine learning and quantum algorithms applied on the same sample with
dimensionality reduction techniques.

| Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
KNN | 72.68 (12.47) | 39.64 (7.18) | 51.07 (8.42) | 43.15 (10.07) | 67.16 (4.12)
CART | 51.43 (10.02) | 52.18 (8.99) | 51.46 (8.86) | 34.59 (10.84) | 67.36 (5.26)
NB | 24.82 (8.84) |  82.45 (28.21) | 38.11 (13.41) | 9.61 (9.35) | 53.78 (3.82)
SVM | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
QSVC |  85.02 (11.42) | 39.24 (8.53) | 52.94 (8.19) |  49.55 (7.54) | 68.45 (3.97)
VQA | 62.50 | 72.22 | 60.61 | 26.57 | 74.09
Table 10: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA) applied to the fraud (bank) detection dataset using the SVD dimensionality reduction. | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 83.63 (9.04) | 64.7 (6.18) |  72.48 (4.73) |  66.24 (5.72) | 80.29 (2.94)
KNN | 80.37 (7.02) | 61.57 (8.58) | 69.5 (7.18) | 62.33 (8.6) | 78.42 (4.75)
CART | 70.76 (9.29) | 73.34 (8.35) | 71.58 (6.67) | 62.54 (8.03) |  81.76 (4.28)
NB | 30.41 (4.79) |  93.01 (3.43) | 45.63 (5.65) | 22.64 (5.44) | 61.36 (3.5)
SVM |  93.20 (5.19) | 55.0 (10.32) | 68.47 (7.92) | 65.19 (7.71) | 76.78 (5.01)
QSVC | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 0.0 (0.0) | 50.0 (0.0)
VQA | 67.39 | 25.41 | 36.90 | 7.16 | 53.09
Table 11: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA) applied to the fraud (bank) detection dataset using the PCA dimensionality reduction. | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR | 87.16 (5.86) | 71.47 (6.47) | 78.18 (3.27) | 72.25 (3.98) | 83.79 (2.77)
KNN | 88.87 (4.79) |  86.52 (4.37) |  87.52 (2.6) |  83.21 (3.65) |  91.23 (1.95)
CART | 78.31 (6.43) | 79.63 (3.65) | 78.88 (4.53) | 71.21 (6.25) | 85.85 (2.84)
NB |  90.75 (7.93) | 58.3 (8.13) | 70.66 (7.21) | 65.76 (8.64) | 78.05 (4.43)
SVM | 88.15 (5.97) | 82.49 (4.81) | 85.05 (3.89) | 80.06 (5.21) | 89.16 (2.53)
QSVC | 56.28 (11.17) | 46.46 (7.21) | 50.3 (6.8) | 35.53 (8.7) | 66.65 (4.02)
VQA | 89.86 | 68.89 | 77.99 | 70.67 | 82.60
Table 12: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC, VQA) applied to the fraud (bank) detection dataset using the SKPP dimensionality reduction.  | Precision (%) | Recall (%) | f1-score (%) | Matthews | Balanced
---|---|---|---|---|---
|  |  |  | corcorref (%) | Accuracy (%)
LR |  94.74 (6.39) | 58.67 (7.57) | 72.14 (6.06) |  68.9 (6.38) | 78.81 (3.84)
KNN | 82.17 (9.5) | 69.7 (8.97) | 74.71 (5.57) | 68.33 (6.78) | 82.34 (4.13)
CART | 70.56 (10.19) | 72.22 (8.58) | 70.77 (6.53) | 61.46 (8.82) | 80.96 (4.49)
NB |  94.74 (6.32) | 58.32 (7.72) | 71.91 (6.43) | 68.64 (7.0) | 78.63 (4.02)
SVM |  94.74 (6.39) | 57.76 (8.17) | 71.36 (6.38) | 68.22 (6.59) | 78.36 (4.08)
QSVC | 82.35 (10.29) | 65.92 (8.79) | 72.93 (8.14) | 66.35 (9.9) | 80.67 (4.94)
VQA | 84.00 | 84.44 | 75.68 | 55.81 | 83.92
Table 13: Classical (LR, KNN, CART, NB, and SVM) and quantum models (QSVC,
VQA) applied to the fraud (bank) detection dataset using the LDA
dimensionality reduction.