Design and analysis of quantum machine learning: a survey

Abstract

Machine learning has demonstrated tremendous potential in solving real-world problems. However, with the exponential growth of data amount and the increase of model complexity, the processing efficiency of machine learning declines rapidly. Meanwhile, the emergence of quantum computing has given rise to quantum machine learning, which relies on superposition and entanglement, exhibiting exponential optimisation compared to traditional machine learning. Therefore, in the paper, we survey the basic concepts, algorithms, applications and challenges of quantum machine learning. Concretely, we first review the basic concepts of quantum computing including qubit, quantum gates, quantum entanglement, etc.. Secondly, we in-depth discuss 5 quantum machine learning algorithms of quantum support vector machine, quantum neural network, quantum k-nearest neighbour, quantum principal component analysis and quantum k-Means algorithm. Thirdly, we conduct discussions on the applications of quantum machine learning in image recognition, drug efficacy prediction and cybersecurity. Finally, we summarise the challenges of quantum machine learning consisting of algorithm design, hardware limitations, data encoding, quantum landscapes, noise and decoherence.

KEYWORDS:

• Machine learning
• quantum computing
• quantum entanglement
• quantum machine learning
• quantum neural networks

1. Introduction

Machine learning has made remarkable progress recently, mainly to the availability of large-scale datasets and the advent of powerful computing systems. Machine learning stands as a fundamental concept within the realm of artificial intelligence. Its objective involves creating systems that can autonomously learn from data, thereby facilitating decision-making with minimal reliance on human input (Liu et al., 2023; Zhang et al., 2022). It has emerged as a focal point of research in both academic and industrial domains, with extensive applications in important fields such as image recognition, public safety (Zhang et al., 2019), healthcare (Garg & Mago, 2021), intelligent transportation (Tian et al., 2015), and natural language processing.

However, there is a rising need for more effective and potent computer systems as the complexity and size of problems continue to expand. The International Data Corporation (IDC) recently published a report that projects a rise in worldwide data generation from 33ZB in 2018 to 175ZB in 2025, with a daily average output of roughly 491EB. As a result, improving computational speed and enhancing storage capacity have become long-term and pressing tasks. Traditional computers face many challenges, such as the nearing of the physical limits of Moore's Law (Waldrop, 2016), difficulty in increasing clock speeds, performance bottlenecks due to the growth of data, and the challenge of handling high-dimensional complex data (Deng et al., 2015). These limitations have prompted the exploration of new computing paradigms, with quantum computing widely considered to have the potential to break through these hardware constraints.

Richard Feynman laid the theoretical groundwork for quantum computing in his paper (Feynman, 1999). He discussed the concept of a computer capable of simulating quantum physical phenomena. Subsequently, Ignacio Cirac and Peter Zoller proposed an approach utilising trapped cold ions in conjunction with laser beams to facilitate quantum processing in their study (Cirac & Zoller, 1995). This study provided detailed plans for the realisation of quantum computers. Since then, quantum hardware has advanced significantly. In the foreseeable future, the realisation of quantum computers with approximately 1000 qubits is expected. Notably, these strides in quantum computing have spurred the advancement of quantum algorithms, resulting in substantial breakthroughs, with key among these are Shor's algorithm (Shor, 1995) and Grover's algorithm (Grover, 1997), both showcasing the powerful performance of quantum algorithms.

Recent years have witnessed a growing body of research that merges quantum computing with machine learning (Adcock et al., 2015; Schuld et al., 2014; Wittek, 2014). Researchers in this field primarily focus on two aspects: Firstly, they are exploring how quantum computing can overcome efficiency challenges in machine learning by utilising parallel processing and exploiting quantum entanglement to boost computational effectiveness. Second, exploring the fusion of quantum mechanics with machine learning principles, the intent is to enhance algorithmic capabilities through the strengths of quantum computing. This integration, likened to the assimilation of other classical theories into quantum physics, has given rise to the field of Quantum Machine Learning (QML). This unique field is dedicated to pushing the boundaries of data analysis dictated by the laws of physics (Biamonte et al., 2017; Wiebe et al., 2014b).

Researchers have discovered that quantum computing methods have shown promising results in various areas of machine learning, such as supervised learning, unsupervised learning, and more. Consequently, they have proposed quantum supervised algorithms, quantum unsupervised algorithms, quantum neural networks, and so on.

In quantum supervised learning, the notion of quantum machine learning took root in 1995, with the inception of quantum neural networks (Huang et al., 2018). Kak et al. introduced the notion of quantum neural computation (Kak, 1995). In the year 2022, Abel and his team introduced the concept of the Perfect Quantum Neural Network. They detailed the method of embedding and training general neural networks within a quantum annealer, entirely avoiding the introduction of classical elements during the training process (Abel et al., 2022). In 2023, Zhou and the team unveiled an innovative quantum neural network model, emphasising “soft quantum neurons.” These soft quantum neurons are fundamental units of soft quantum computing, influenced solely by single-qubit operations and measurements under classical control. This approach significantly reduces the complexity of quantum implementation (Zhou, Liu, et al., 2023). Additionally, Wiebe and colleagues put forward gradient estimation techniques relying on quantum sampling and quantum amplitude estimation (Wiebe et al., 2014b). In addition to quantum neural networks, there have also been breakthroughs in other aspects of quantum supervised learning. In 2014, Lu and colleagues introduced a quantum decision tree classifier, along with the quantum entropy impurity criterion for deciding which nodes should undergo splitting (Lu & Braunstein, 2014). In 2019, Khadiev put forward a quantum variant of the C5.0 algorithm, where the decision tree classifier created by the C5.0 algorithm relies on an extension of the Grover search algorithm (Khadiev et al., 2019). Rebentrost et al. Introduced Quantum Support Vector Machines (QSVM) in 2014 and provided an example of a quantum “big data” algorithm (Rebentrost et al., 2014). In 2022, Zhang and his team unveiled a quantum support vector machine that utilises the regularised Newton method. This innovation surmounted the constraints encountered in applying the HHL algorithm within Rebentrost's version of SVM, particularly when dealing with inadequately structured input matrices (Zhang et al., 2022). Moving to 2023, Golchha and colleagues introduced a greyscale image binary classification model that relies on the quantum support vector classifier (Golchha & Verma, 2023). A quantum adaptation of the k-nearest neighbour algorithm was introduced by Wiebe et al. (Wiebe et al., 2014a). This quantum version enables the determination of nearest neighbours by assessing the Euclidean distance between data points, eliminating the necessity for measurement. In 2022, Quezada and colleagues introduced a quantum k-nearest neighbours algorithm through the development of a quantum sorting algorithm (Quezada et al., 2022).

In the field of quantum unsupervised learning, Lloyd et al. introduced the Quantum k-Means algorithm, applying the superposition property of quantum states to classical vector representations. This algorithm has the theoretical potential for efficient clustering of massive data sets (Lloyd et al., 2013a). Hou and colleagues in 2022 introduced an approach that combines fuzzy theory, quantum computing, and the k-Means algorithm, broadening the applicability and improving the accuracy of the algorithm (Hou et al., 2022). In 2013, Lloyd and his team proposed quantum principal component analysis, utilising Gram matrices, which are covariance matrices of a set of vectors, to represent the density matrices of quantum systems because they are Hermitian matrices (Lloyd et al., 2013b). In 2021, Li introduced resonant quantum principal component analysis and developed a harmonic analysis algorithm that only requires a frequency scan to extract principal components (Li, Chai, et al., 2021). While the core ideas of these quantum machine learning algorithms are similar to traditional algorithms, their main distinction lies in harnessing the high parallelism of quantum computation to address computationally expensive sub-steps (Huang et al., 2018). Based on the above descriptions of quantum supervised and unsupervised learning, the quantum versions of various machine learning algorithms are depicted in Table 1.

Table 1. Quantum version of machine learning.

Method	Ref	Remarks
Neural Network	(Abel et al., 2022)	The authors proposed embedding a general neural network into a quantum annealer, conducting quantum-based training by converting the loss function into an Ising model Hamiltonian.
	(Zhou, Liu, et al., 2023)	The authors presented a quantum neural network using “soft quantum neurons,” reducing exponential state space growth through single-qubit operations and classical control.
Decision Tree	(Lu & Braunstein, 2014)	The authors introduced a quantum decision tree model using von Neumann entropy and quantum clustering algorithms to classify quantum objects, bridging quantum computing and machine learning.
	(Khadiev et al., 2019)	The authors introduced a decision tree classifier created with the C5.0 algorithm, which relies on a quantum subroutine based on the Grover algorithm.
Support vector machine	(Zhang et al., 2022)	The authors introduced an RN-QSVM (Quantum Support Vector Machine using Regularized Newton) that provides exponential speedup by addressing the input matrix's constraints.
	(Golchha & Verma, 2023)	The authors introduced a model based on a quantum support vector classifier for binary greyscale image classification, achieving improved classification accuracy through preprocessing and quantum techniques, surpassing classical support vector machines.
k-nearest neighbours	(Quezada et al., 2022)	The authors presented a quantum sorting algorithm that offers flexibility in memory and circuit depth requirements. They subsequently applied this algorithm to create a quantum version of a classical machine learning algorithm.
k-Means	(Hou et al., 2022)	The authors presented the pioneering Quantum Fuzzy k-Means algorithm. This streamlines sample classification with improved efficiency and accuracy.
Principal component	(Li, Chai, et al., 2021)	The authors have developed a harmonic analysis algorithm with minimal auxiliary quantum bit resources for extracting principal components of unknown low-rank density matrices.

Quantum machine learning has been a subject of ongoing exploration in research for a considerable period. Its aim is to leverage quantum encoding, quantum entanglement, and superposition in quantum computing to improve traditional machine learning. In practice, evidence substantiates that quantum algorithms can markedly outperform classical algorithms, sometimes even exponentially faster. The comparison of Quantum Algorithm Time Complexity and Classical Algorithm Time Complexity is depicted in Table 2.

Table 2. Comparison of Quantum Algorithms.

	Time Complexity
Method	QML	ML	References
k-nearest Neighbors	O(log(n·m))	O(n·m)	(Wiebe et al., 2014a)
Principle Component Analysis	O(log(d))	O(d)	(Lloyd et al., 2013b)
Deep Learning (Amplitude Estimation)	$O\left(\sqrt{N}\cdot E^2\cdot \left(\sqrt{k}+\underset{x}{max}\sqrt{k_x}\right)\right)$	O(N·E·k)	(Wiebe et al., 2014b)
k-Means	O(k·log(k·m·n))	O(m·n)	(Lloyd et al., 2013a)
SVM	O(log(m·n))	O(m·n)	(Li et al., 2015)

Though there have been significant strides in quantum technology, challenges endure in quantum machine learning. Specifically, the pursuit of a fault-tolerant, universal quantum computer with a considerable number of qubits remains a key challenge. It remains unclear how many logical qubits a quantum computer would need to surpass the computational power of even the most powerful classical computers (Adcock et al., 2015). On the other hand, the current state of quantum hardware, in terms of quality, speed, and scalability, falls short of fully realising the potential of quantum computation, necessitating improvements in quantum hardware (Garcia et al., 2022). Currently, the development of universal large-scale quantum computers is ongoing, while research on algorithms has made some advancements.

Building upon the contextual groundwork provided for quantum machine learning, we will discuss Design and Analysis of Quantum Machine Learning. To be precise, the primary highlights of our contributions are outlined as follows:

• We introduced the relevant concepts of quantum computing to provide readers with a preliminary understanding. Then, in-depth discussions were conducted on quantum support vector machines, quantum neural networks, quantum k-nearest neighbour algorithms, quantum principal component analysis, and quantum k-Means algorithms in quantum machine learning.

• Quantum machine learning boasts a multitude of practical applications across various scenarios. In this study, we mainly introduce three application scenarios, namely image recognition, drug prediction, and network security, and conduct in-depth discussions.

• With the continuous in-depth exploration of quantum machine learning, the challenges in this field have also been noticed. In this study, we mainly discuss five challenges, such as quantum machine learning algorithm design, quantum hardware limitations, quantum dataset and encoding, and quantum noise and decoherence.

Our objective in this paper is to present the fusion of quantum computing within the field of machine learning, encompassing the entire spectrum from foundational principles to practical applications. In Section 2, we introduced classical machine learning, covering three aspects. Subsequently, we provided an overview of quantum computing concepts and introduced quantum algorithms. In Section 3, we embark on an exploration of key quantum machine learning algorithms that yield enhanced performance compared to their conventional counterparts. Section 4 delves deeply into the application scenarios where quantum machine learning exhibits its potential. In Section 5, the challenges currently faced by quantum machine learning are discussed. Finally, Section 6 serves as a comprehensive summary of this article. The overall structure of this article is illustrated in Figure 1.

Figure 1. The overall idea of this paper.

2. Background

2.1. Classical machine learning

A field within artificial intelligence, referred to as “machine learning,” aims to enable computers to learn from data and evolve independently, without the need for direct programming guidance (Chen et al., 2023). Building and training models are required to enable computers to recognise patterns, anticipate outcomes, and make decisions based on data, gradually enhancing their performance (Zhang, He, et al., 2024; Zhang, Hu, et al., 2024). Machine Learning is applied across diverse domains and industries. Its prevalence spans a multitude of fields, encompassing speech and image recognition, recommendation systems, financial projections, medical diagnostics, and natural language processing.

Various learning methodologies and task types allow for a broad categorisation of machine learning into several categories:

• Supervised Learning: With labelled training data, each sample in supervised learning has a predetermined label or category. The algorithm's objective is to discover a function that converts input data into the appropriate labels. Classification (e.g. picture classification, spam detection) and regression (e.g. house price prediction, stock price prediction) are frequent supervised learning tasks.

• Unsupervised Learning: Unsupervised learning involves using unlabelled training data, where known classes or labels are absent. The algorithm's goal is to uncover latent structures or patterns within the data, encompassing tasks such as data clustering, dimensionality reduction, or anomaly detection. Typical unsupervised learning tasks include clustering (e.g. user segmentation, image segmentation) and dimensionality reduction (e.g. principal component analysis).

• Reinforcement Learning: Reinforcement learning involves the dynamic interplay between an agent and its surroundings, where optimal behaviour is learned through a process of trial and error, guided by reward mechanisms. It is suitable for problems that require decision-making in dynamic environments, such as playing games like Go or autonomous driving.

A successful algorithm has two crucial advantages. Firstly, it can replace labour-intensive and repetitive manual work. Secondly, and more importantly, it can discover more complex and subtle patterns from input data than an ordinary human observer can (Naqa & Murphy, 2015).

2.2. Quantum computing related concepts

Quantum computing operates by leveraging laws derived from quantum physics to process data, utilising phenomena like superposition, entanglement, and quantum measurement. Superposition, a unique characteristic of quantum mechanics, allows objects to simultaneously exist in multiple states. Entanglement enables the correlation of two or more quantum particles, even across significant spatial separations. The conversion of quantum information into classical information is referred to as quantum measurement (Zhou et al., 2014). This section delves into the fundamentals of quantum computing, encompassing aspects ranging from quantum bits to quantum algorithms.

2.2.1. Qubit

Dirac(Bra-Ket)Notation—Dirac notation is a symbolic system introduced by the British physicist Paul Dirac for representing states and operators in quantum mechanics. In the field of quantum mechanics, we use Dirac notation to express quantum states and operators. This notation is based on the concepts of vectors and inner products, and it introduces Ket and Bra symbols.

Let a and b be in C², where Ket and Bra are correspondingly as $|a〉$and $〈b|$ in the following. (1) $|a〉=\left(\begin{array}{l}{a}_1\\ a_2\end{array}\right),〈b|=|b{〉}^{\ast }={\left(\begin{array}{l}{b}_1\\ b_2\end{array}\right)}^{\ast }=\left(\begin{array}{cc}{b}_1^{\ast }& b_2^{\ast }\end{array}\right).$(1) It is important to mention that adjusting the imaginary part of a complex number leads to deriving its complex conjugate. For instance, if we have a complex number b = a + i × d, its complex conjugate is represented as b* = a - i × d.

Bra-Ket: Inner product (2) $〈b|a〉=a_1{b}_1^{\ast }+a_2{b}_2^{\ast }=〈a|b{〉}^{\ast }.$(2) Ket-Bra: Outer product (3) $|a〉〈b|=\left(\begin{array}{cc}{a}_1{b}_1^{\ast }& a_1{b}_2^{\ast }\\ a_2{b}_1^{\ast }& a_2{b}_2^{\ast }\end{array}\right).$(3) In the realm of quantum computing, the foundational unit of information is denoted as a quantum bit, commonly known by its abbreviated form, “qubit.” Think of it as an extension of classical bits from classical computing into the realm of quantum mechanics. Specifically, a qubit represents a two-dimensional quantum system. Typically, qubits are symbolised using Dirac notation: (4) $|0〉=\left(\begin{array}{l}1\\ 0\end{array}\right),|1〉=\left(\begin{array}{l}0\\ 1\end{array}\right).$(4) A qubit may exist in three potential states: $|0$, $|1$, or in a combined superposition of both. (5) $|\psi 〉=\alpha |0〉+\text{β}|1〉=\left(\begin{array}{l}\alpha \\ \text{β}\end{array}\right).$(5) Where $\alpha ,\text{β}\in \mathbb{C}$. In a quantum system, suppose there are two quantum states $|0〉$and $|1〉$, simultaneously in a superposition state $|\psi 〉$. The parameters $\alpha$ and $\text{β}$ are complex numbers that represent the coefficients of this superposition state. They satisfy the condition $|\alpha {|}^2+|\text{β}{|}^2=1$, which means the sum of the squares of their magnitudes is equal to 1. Upon conducting a measurement on this quantum system, it undergoes a collapse into one of two potential states. The probability of yielding the measurement outcome $|0〉$ is represented by $|\alpha {|}^2$, while the probability of yielding the measurement outcome $|1〉$corresponds to $|\text{β}{|}^2$. This indicates that before the measurement, we cannot determine in which state the quantum system exists, and the measurement outcome is determined probabilistically according to the coefficients of the superposition state. In other words, the measurement result is random, but the probabilities are influenced by the coefficients of the superposition state.

2.2.2. Quantum gates

Quantum gates, essential operations in quantum computing, can alter the states of quantum bits (qubits). While they bear a resemblance to classical logic gates, quantum gates possess unique attributes exclusive to quantum computing (Wan et al., 2014; Wu et al., 2019). By using quantum gates, we can achieve operations such as state flipping, information exchange, and the creation of entangled states to accomplish various quantum computing tasks. Quantum gates serve as crucial elements in constructing quantum circuits and executing quantum algorithms, as they provide the foundation for harnessing the advantages of quantum computing. Quantum gates, symbolised through unitary matrices, maintain the normalisation and reversibility of quantum states. They operate as transformers, transitioning one quantum state into another while ensuring an equivalent count of input and output qubits. The amalgamation of various quantum gates facilitates the construction of intricate quantum circuits, serving a diverse spectrum of quantum computing objectives and the enhancement of quantum algorithms. The conversion from one quantum state to another, orchestrated by quantum gates, can be depicted as: (6) $U|\psi 〉=\left(\begin{array}{cc}{u}_{11}& u_{12}\\ u_{21}& u_{22}\end{array}\right)\left(\begin{array}{c}\alpha \\ \text{β}\end{array}\right)\text{ = }\left(\begin{array}{c}\text{a}\\ b\end{array}\right)=|\phi 〉.$(6) Table 3 lists some basic quantum gates.

Table 3. Basic quantum gates.

	Quantum Logic Gate Name	Matrix Representation	Quantum circuit symbol	Quantum Gate Function
Elementary quantum gates	Pauli-X	$X=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\equiv {\sigma }_x$		The gate circuit maps the quantum state $|0〉$ to $|1〉$ and the quantum state $|1〉$ to $|0〉$, which is equivalent to a NOT gate.
	Pauli-Y	$Y=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)\equiv {\sigma }_y$		The gate circuit maps the quantum state $|0〉$ to $i|1〉$ and the quantum state $|1〉$ to $-i|0〉$.
	Pauli-Z	$Z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\equiv {\sigma }_z$		This gate circuit maintains the integrity of the basis state $|0〉$ while inverting the phase of the state $|1〉$, resulting in $-|1〉$. It is commonly known as the phase-flip gate.
	Hadamard	$H={\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$		Mapping $|0〉$ to $\frac{|0〉+|1〉}{\sqrt{2}}$ and $|1〉$ to $\frac{|0〉-|1〉}{\sqrt{2}}$ results in a superposition.
Two-qubit gates	Controlled NOT	$\mathrm{CNOT}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)$		It operates on two qubits, and only when the control qubit is $|1〉$ does it perform the NOT operation on the target qubit.
	SWAP	$\mathrm{SWAP}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$		This gate circuit is used to operate on two qubits and swap the states of these two qubits.

2.2.3. Quantum measurement

Much like how classical computers operate, quantum computers perform particular computational tasks by running quantum circuits which are comprised of a series of quantum gates. The final step within a quantum circuit involves quantum measurement. which permits the transition of information from the quantum realm to the classical realm (Li, 2022).

A set of measurement operators {Mm}, acting within the quantum state space $|0$, can imitate the quantum measurement procedure. Here, “m” represents the feasible outcomes of the measurement. In the context of a single measurement, the result is inherently uncertain, and the likelihood of obtaining outcome “m” is mathematically determined by: (7) $p\left(m\right)=〈\psi |M_m^{+}{M}_m|\psi 〉,$(7) and following the measurement, the system's state is: (8) $\frac{M_m|\psi 〉}{\sqrt{〈\psi |M_m^{+}{M}_m|\psi 〉}}.$(8) The completeness equation is satisfied by the measurement operators: (9) $\sum _m{M}_m^{+}{M}_m=I.$(9)

2.2.4. Quantum entanglement

Regardless of the distance separating quantum particles, a phenomenon known as quantum entanglement causes them to interact and display correlations. When entangled particles are measured, even though they are far away, the state of one particle immediately impacts the state of the other. For instance, if one particle exhibits a “down” spin upon measurement, the other particle will instantly exhibit an “up” spin. Regardless of where the particles are in the cosmos, this entanglement is still present. This is an important illustration of how wave-particle duality permits qubits to interact and occasionally interfere with quantum computations. We examine, for instance, the Bell state$\frac{1}{\sqrt{2}}(|00〉+|11〉)$, which is equally likely to be measured in the states $|00〉$ and $|11〉$, with a 50% chance of each. The superposition collapses when one of them measures something, seemingly immediately affecting the other (Zeguendry et al., 2023).

The existence of entangled states is a significant physical phenomenon that has a profound impact on quantum computing and its information processing (Kang et al., 2019). In fact, without entangled states, quantum computers would lack the capability to exhibit superior computational power compared to classical computers.

2.3. Quantum algorithms

A quantum algorithm is a distinct form of algorithm engineered to harness the potential of quantum computers and the foundational principles of quantum mechanics to solve problems. In contrast to conventional classical algorithms, quantum algorithms tap into the distinct computational potency of quantum bits (qubits), utilising attributes like superposition, entanglement, and quantum coherence. By leveraging the quantum states and quantum gate operations of qubits based on principles of quantum mechanics, quantum computing achieves exponential computational speedup in certain problems. It is essential to ensure that quantum algorithms, exhibit greater efficiency compared to their counterparts tailored for conventional computing systems. This can be achieved by evaluating the complexity of the task using various methods, such as exponential, polynomial, logarithmic, or linear scaling of computational time, or a combination of these methods. Through this evaluation, one can verify if the quantum algorithm surpasses the efficiency of traditional algorithms (Zeguendry et al., 2023). In this section, we will present a selection of notable quantum algorithms, including the HHL algorithm, the Grover algorithm, and the Deutsch-Jozsa algorithm.

2.3.1. HHL algorithm

The Harrow-Hassidim-Lloyd (HHL) algorithm, initially proposed by Harrow et al. in 2008 (Harrow et al., 2008). It is a quantum computing algorithm designed for the resolution of linear systems of equations. Conventional methods for solving linear systems typically have exponential time complexity, while the HHL algorithm leverages the parallelism and superposition properties of quantum computing to solve specific linear systems in polynomial time.

The HHL algorithm comprises three key components: phase estimation, controlled rotation, and inverse phase estimation. These are the general steps:

• Quantum phase estimation (Wossnig et al., 2018): This subroutine applies the phase estimation technique to the initial qubits, which is a fundamental step in decomposing quantum states into specific bases. We initialise the quantum state $|b〉$. At this point, the initial state of the entire system is represented as $|0{〉}^{\otimes n}|b〉$. Upon completion of the phase estimation subroutine, the state of the entire system becomes $\sum _j{\text{β}}_j|{\lambda }_j〉|{\mu }_j〉$.

• Controlled rotation. This operation applies a controlled phase shift to the state $|{\lambda }_j〉$, resulting in the transformation ${{\lambda }_j}^{-1}|{\lambda }_j〉$.

• The application of inverse quantum phase estimation allows the register holding the state $|{\lambda }_j〉$to be reset back to $|0〉$ At this point, the state of the entire system is: (10) $\sum _j(\sqrt{1-\frac{c^2}{{\lambda }_j^2}}|0〉+\frac{c}{{\lambda }_j}|1〉)\text{β}|0〉|{\mu }_j〉.$(10)

The quantum phase estimation module aims to swiftly estimate the phase φ of an eigenvalue $e^{2\mathrm{\pi i\phi }}$ of a unitary matrix. Since the eigenvalues of a unitary matrix are complex numbers with a modulus of 1, for a unitary matrix, the eigenvalues and phases are essentially equivalent. The circuit diagram is depicted in Figure 2.

Figure 2. HHL Algorithm Circuit Diagram.

2.3.2. Deutsch-Jozsa algorithm

The Deutsch-Jozsa algorithm, introduced by David Deutsch and Richard Jozsa in 1992 (Deutsch & Jozsa, 1992), represents one of the earliest and simplest quantum algorithms showcasing quantum computing's superiority over classical algorithms. This algorithm addresses the “balanced versus constant” problem, which involves determining if a given function f(x), where x is a bit string, consistently produces either 0 or 1. The primary objective of the algorithm is to determine the properties of function—whether it is constant or balanced without checking all possible inputs x.

Suppose $f:\{0,1{\}}^{\otimes n}\to \left\{0,1\right\}$ is a function, and n + 1 quantum bits are prepared in the state $|0{〉}^{\otimes n}|1〉$, where the first n quantum bits are in the state $|0〉$ and the last target quantum bit is in the state $|1〉$ (Nagata et al., 2017). Figure 3 shows the “black box” circuit of the Deutsch-Jozsa algorithm. In the circuit, H represents the Hadamard gate, which acts as $H|0〉=\frac{1}{\sqrt{2}}(|0〉+|1〉)$ and $H|1〉=\frac{1}{\sqrt{2}}(|0〉-|1〉)$. U_f denotes the Oracle circuit.

Figure 3. Schematic diagram of Deutsch-Jozsa circuit.

The implementation steps of the Deutsch-Jozsa algorithm, starting with the initial state $|{\phi }_0〉=|0{〉}^{\otimes n}|1〉$, are as follows:

Step 1: Implement a series of n + 1 Hadamard gates on each quantum bit separately, yielding the ensuing state: $|{\phi }_1〉=\sum _{x=0}^{2^n-1}\frac{|x〉}{\sqrt{2^n}}\frac{|0〉-|1〉}{\sqrt{2}}$.

Step 2: Apply U_f to these n + 1 quantum bits, resulting in the following state:$|{\phi }_2〉=\sum _{x=0}^{2^n-1}\frac{{(-1)}^{f\left(x\right)}}{\sqrt{2^n}}|x〉\left(\frac{|0〉-|1〉}{\sqrt{2}}\right)$.

Step 3: Operate n Hadamard gates on each of the first n quantum bits separately, resulting in the following state:$|{\phi }_3〉=\frac{1}{2^n}\sum _{x=0}^{2^n-1}\sum _{z=0}^{2^n-1}{(-1)}^{f\left(x\right)+\mathrm{xz}}|z〉\left(\frac{|0〉-|1〉}{\sqrt{2}}\right)$.

Step 4: Perform a measurement on the first n quantum bits, considering only the probability of obtaining the result $|0{〉}^{\otimes n}$. This is done by substituting $|z〉=|0〉$ into $|{\phi }_3〉$, where the coefficient is given by$\frac{1}{2^n}\sum _{x=0}^{2^n-1}{(-1)}^{f\left(x\right)+\mathrm{xz}}|{}_{z=0}=\frac{1}{2^n}\sum _{x=0}^{2^n-1}{(-1)}^{f\left(x\right)}$.

From the above derivation, it can be seen that when the function f(x) is constant, we will always obtain the measurement result $|0{〉}^{\otimes n}$. On the other hand, when the function f(x) is balanced, we will never obtain the measurement result $|0{〉}^{\otimes n}$. Conversely, by observing the first n quantum bits, we are equipped to determine if the function f(x) maintains a constant state or is balanced (Gruska et al., 2014).

2.3.3. Grover’s algorithm

Grover’s algorithm, introduced in 1996, is a well-known quantum algorithm designed to accelerate database searches in the quantum realm. Similar to the Deutsch-Josza algorithm, Grover’s algorithm navigates through a set of elements to identify those that meet a particular criterion. Compared to traditional sequential searching, it leverages the advantage of quantum superposition to accelerate the search process. Grover’s algorithm offers significant performance improvements compared to an unsorted database. This method relies on an “oracle” or “black-box” function to assess whether a specific set of input quantum bits complies with the search criterion. For instance, this method can be employed to verify whether a given phone number exists within a phone number list. The black-box function compares the provided phone number with those in the list, yielding a 1 for a match and a 0 for non-matching entries. This black-box function acts as a quantum oracle, capable of assessing 2N quantum bit registers concurrently. Thus, it has the property of outputting 1 at the designated location and 0 elsewhere. The algorithm’s computation time depends on the square root of the list size, as opposed to the linear time of traditional methods. This provides an interesting improvement from linear to square root time. Furthermore, Grover’s algorithm can be integrated into other algorithms, such as identifying the shortest or longest path through a graph or determining the minimum or maximum number among a group of N numbers.

3. Quantum machine learning

Quantum machine learning harnesses the intrinsic parallel processing capabilities of quantum computing, elevating the performance of traditional machine learning methods. Its core principle involves translating information from classical learning algorithms into quantum states and conducting unitary evolutions on these states for computational purposes. Only classical learning steps that satisfy the conditions for quantum computing can be accelerated (Chen et al., 2016; Chen et al., 2019; Chen et al., 2021). To exploit quantum properties for acceleration, it is necessary to appropriately modify traditional algorithms to meet the requirements of quantum computing. Current research primarily focuses on replacing specific sub-processes of classical algorithms with quantum algorithms to decrease computing complexity and increase algorithm efficiency (Cao et al., 2017; Cao et al., 2019; Cao et al., 2020). Fundamentally, quantum machine learning unites the realms of quantum computing and traditional machine learning. By categorising data sources as either classical (C) or quantum (Q) and distinguishing the computational device for data processing as classical (C) or quantum (Q), the fusion of machine learning and quantum computing can be approached through four distinctive strategies, as depicted in Figure 4 (Jadhav et al., 2023).

Figure 4. Four distinct approaches to combining quantum computing and machine learning.

The four types of integrated scenarios include classic-classic (CC) methods, classic-quantum (CQ) methods, quantum-classic (QC) methods, and quantum-quantum (QQ) methods. These methods are used for information transfer and processing between classical computing and quantum computing. Here’s a brief introduction to each method:

Classic-classic (CC) method: This is the traditional classical computing method that uses classical hardware and algorithms for information processing. This method is suitable for general computing tasks on classical computers, such as numerical calculations, data processing, and algorithm execution.

Classic-quantum (CQ) method: This method leverages quantum computing to accelerate classical machine learning algorithms, combining quantum and classical approaches to achieve higher performance, In this situation, classical computers are commonly used to manage certain aspects of computation, while quantum computers handle specific tasks that can benefit from quantum parallelism, superposition, and entanglement (Abohashima et al., 2020). For instance, Adhikary and colleagues introduced a novel variational quantum classifier that encodes high-dimensional data using quantum systems, achieving high accuracy with a single training run (Adhikary et al., 2020). On the other hand, Havlicek and his team proposed two models for subspace support vector machines, based on quantum variational circuits and quantum kernel estimation (Havlíček et al., 2019). Schuld et al. put forward hybrid quantum techniques to address classification problems (Schuld & Killoran, 2019), while Mitarai and co-authors introduced hybrid quantum-classical techniques for tasks like classification, regression, and clustering (Mitarai et al., 2018).

Quantum-classic (QC) method: This technique entails executing algorithms inspired by quantum concepts on conventional, classical computers (Buffoni & Caruso, 2021). For instance, Ding and colleagues have drawn inspiration from quantum support vector machines (SVM) to propose a new algorithm for addressing exponentially accelerated classification problems. This algorithm is based on linear transformations to enhance the classification speed (Ding et al., 2019). Sergioli and their research team introduced a novel binary supervised learning classifier, drawing inspiration from quantum principles, termed the Helstrom Quantum Centroid (HQC). This model is built on density matrices and the fundamental tenets of quantum theory (Sergioli et al., 2019).

Quantum-quantum (QQ) method: This method involves using multiple quantum systems for information processing and quantum operations. Its applications span quantum communication, algorithms, and simulation, enabling the execution of intricate and more efficient computational tasks in the quantum realm.

These five aspects in quantum computing—Quantum Support Vector Machine (QSVM), Quantum Neural Network (QNN), Quantum k-Nearest Neighbors Algorithm (QKNN), Quantum Principal Component Analysis (QPCA), and Quantum k-Means Algorithm—are worth exploring. Let's delve into each, examining their applications and significance within the quantum computing domain.

3.1. Quantum support vector machine

The quantum SVM was proposed by Rebenentrost et al. in 2014, and over the past 10 years, it has made significant progress (Allcock & Hsieh, 2020; Havenstein et al., 2018; Kerenidis et al., 2019; Li et al., 2015; Willsch et al., 2020; Ye et al., 2020). Various techniques have been combined to improve the QSVM, including the least-squares quantum support vector machine(LS-QSVM), the QSVM modified by the Grover algorithm, the QSVM enhanced by the HHL algorithm, as well as the recently studied quantum kernel support vector machine (QKSVM) and quantum variational support vector machine (QV-SVM) by INNAN et al. Additionally, the combination of these techniques has led to the development of the quantum variational kernel support vector machine (QVK-SVM) (Innan et al., 2023).

In this paper, we will summarise and explain the least-squares approximation of quantum support vector machines (QSVM) (Zhang & Ni, 2020).

The representation of the normal vector $\overrightarrow{w}$ is as follows: (11) $\overrightarrow{w}=\sum _{i=1}^N{\alpha }_i\overrightarrow{x_i}.$(11) where ${\alpha }_i$ the weight of the i-th training vector$\overrightarrow{x_i}$. The main concept of this study is to optimise the parameters ${\alpha }_i$ and b by transforming them into a least squares reconstruction support vector machine (SVM) framework. Instead of using traditional quadratic programming methods, this approach bypasses the quadratic programming step and solves a system of linear equations to obtain the optimal parameter values. By doing so, it allows for a more efficient parameter estimation process and reduces the computational complexity during optimisation. This approach offers a practical substitute for parameter optimisation: (12) $F\left(\begin{array}{c}b\\ \overrightarrow{\alpha }\end{array}\right)\equiv \left(\begin{array}{cc}0& 1\\ I& K+{\gamma }^{-1}{I}_N\end{array}\right)\left(\begin{array}{l}b\\ \overrightarrow{\alpha }\end{array}\right)=\left(\begin{array}{l}0\\ \overrightarrow{y}\end{array}\right),$(12) In this case, K represents the linear kernel matrix, which is defined as $K_{i,j}=\overrightarrow{x_i}\cdot \overrightarrow{x_j}$. Initially, through the utilisation of the training-data oracle, the conventional training data undergoes encoding as: (13) $|\overrightarrow{x}〉=\frac{1}{|\overrightarrow{x_i}|}\sum _{i=1}^M(\overrightarrow{x_i}{)}_j|j〉.$(13) We can prepare the following state from the initial state $\left(1/\sqrt{M}\right)\sum _{i=1}^M|i〉$ and the training-data oracle: (14) $|s〉=\frac{1}{\sqrt{N_S}}\sum _{i=1}^M|\overrightarrow{x_i}||i〉|\overrightarrow{x_i}〉.$(14) We can use the HHL algorithm to resolve the linear equation system in order to optimise the hyperplane parameters b and a_i. Matrix inversion is used to get the hyperplane parameters: (15) $(b,{\overrightarrow{a}}^T{)}^T=F^{-1}(0,{\overrightarrow{y}}^T{)}^T.$(15) Because the quantum register is initialised as (16) $|0,y〉=\frac{1}{\sqrt{N_{0,y}}}(|0〉+\sum _{i=1}^M{y}_i|i〉).$(16) It will be possible to transfer quantum states by executing the inverse operation on matrix F. (17) $|b,a〉=\frac{1}{\sqrt{N_{b,a}}}(b|0〉+\sum _{i=1}^M{a}_i|i〉).$(17) The classification outcome then has the following representation with the optimised parameters b and a_i: (18) $y\left(\overrightarrow{x_0}\right)=\mathrm{sgn}\left(\sum _{i=1}^M{a}_i(\overrightarrow{x_i}\cdot \overrightarrow{x_0})+b\right).$(18) Quantum SVM is widely used in fields such as implement handwriting recognition (Li et al., 2015), binary classification of grey scale images (Golchha & Verma, 2023) and implementation of automation in a radio access network (Yang et al., 2019). In particular, for a deeper grasp of QSVM's application, Willsch and colleagues (Willsch et al., 2020) conducted training of Kernel-based support vector machines on a D-Wave 2000Q quantum annealer. Their research results indicate that quantum annealing generates a different set of solutions, often exhibiting better generalisation on invisible data compared to a single global minimum implemented by SVM trained on classical computers.

In the study by Ref. (Willsch et al., 2020), as illustrated in Figure 5, the dataset D encompasses n = 1–40 points (x_n, t_n). This set involves an outer region hosting the negative class t_n = −1 and an inner region representing the positive class t_n = 1. Here, rn is equal to 1 if t_n = −1, and rn is 0.15 if t_n = 1. It was generated according to Based on the findings in Ref. (Willsch et al., 2020), as illustrated in Figure 5, the dataset D comprises n = 1–40 data points (x_n, t_n). This dataset encompasses both the outer region with a negative class t_n = −1 and the inner region with a positive class t_n = 1, where r_n = 1 for t_n = −1 and r_n = 0.15 for t_n = 1. This dataset was generated in accordance with. (19) $X_n=r_n\left(\begin{array}{c}\cos {\phi }_n\\ \sin {\phi }_n\end{array}\right)+\left(\begin{array}{c}{s}_n^x\\ s_n^y\end{array}\right),$(19) In this case, ${\phi }_{\text{n}}$ is evenly distributed within the range [0, 2π) for each class, while $s_n^x$ and $s_n^y$ are sampled from a normal distribution with a mean of 0 and a standard deviation of 0.2.

Figure 5. Visualisation of the classification boundary resulting by the classical SVM (a), and the quantum SVM (b-d) (Willsch et al., 2020).

In Figure 5(a), the decision boundaries for the classical SVM (3, 16) are depicted, while Figures 5(b)–(d) showcase three distinct solutions obtained from the ensemble generated by the quantum SVM (2, 2, 0, 16). It's important to note that the data points displayed do not represent a separate testing set but rather correspond to the same 40 points used for training the SVM models. The varying background colours in each instance signify their proximity to the decision boundary.

The results of Figure 5 have well shown the that, comparing with the classic SVM in Figure 5 (a), however, Figure 5 (b)-(d) automatically generates a range of alternative classifiers. In Figure 5 (b), represented by sample #1, it exhibits similarities to the characteristics of the global minimum. On the other hand, Figure 5 (c) with sample #6 results in a narrower boundary around the outer circle, and Figure 5 (d) with sample #16 even exhibits sensitivity to the gaps in the outer circle. This observation implies that the ensemble of classifiers derived from quantum SVM might possess greater efficacy compared to the individual classifier produced by classical SVM.

3.2. Quantum neural network

A quantum neural network is an innovative model that combines principles of quantum computing with artificial neural network theory. Its aim is to leverage quantum mechanics to enhance the computational efficiency and capabilities of conventional neural networks. In traditional artificial neural networks, data is represented and processed using binary bits. However, in a quantum neural network, the basic informational unit is a qubit, a quantum state. Unlike classical bits, qubits possess the unique ability to exist in a superposition of various states, referred to as a quantum superposition state. The distinctive advantage of these quantum superposition states lies in their capacity for parallel computations, which significantly boosts computational efficiency.

The fundamental and crucial component of Quantum Machine Learning (QML) models is the Parametrized Quantum Circuit (PQC). PQC conceptually resembles a neural network and can formally embed classical neural networks (Thanasilp et al., 2021; Wan et al., 2016). This analogy is accurate, and certain types of PQCs are referred to as Quantum Neural Networks (QNNs). QNNs can be classified into three different types:

• Dissipative QNN (Cerezo et al., 2022): In dissipative QNNs, the prior quantum bits within a layer are discarded once the information transfers to the subsequent layer with new quantum bits. This type of QNN is useful for problems involving time evolution and dissipative processes. By discarding old quantum bits, it simulates the interaction of quantum systems with the environment and the effect of decoherence.

• Standard QNN (Cerezo et al., 2022): Standard QNNs are the most common type of QNN. They utilise quantum circuits to propagate quantum data states, and as the quantum network deepens and branches out, no quantum bits are lost or added. This type of QNN is commonly used for tasks such as classification, regression, and generative modelling.

• Convolutional QNN (Cerezo et al., 2022): Convolutional QNNs are specifically designed to handle high-dimensional data. They utilise convolutional processes to condense the data's dimensionality, ensuring the retention of crucial features. This type of QNN enables data dimensionality reduction within the framework of quantum computation, providing more efficient processing and learning approaches.

At the core of a Quantum Neural Network (QNN) lies the quantum neuron, analogous to the quantum counterpart of an artificial neuron. This quantum neuron receives a quantum state as input, encompassing both amplitude and phase. The output is likewise a quantum state, and the amplitude weights can have complicated values. A perceptron equivalent in this context is described as: (20) $|y〉=U\sum _{n=0}^{n-1}{w}_{\mathrm{ny}}|x_n〉,$(20) where $w_{\mathrm{ny}}$ are the weights and U is the activation function produced by any operator. The learning rule given in (Altaisky et al., 2014) can be used to train the perceptron. (21) $w_{\mathrm{iy}}(t+1)=w_{\mathrm{iy}}\left(t\right)+\eta (|d〉-|y〉)〈x_i|.$(21) where “d” represents the target state, “y” signifies the neuron state at time “t”, and “η” denotes the learning rate.

The existing types of QNN models discovered so far comprise the following: Quantum M-P neural network, Quantum competitive neural network, Quantum-inspired neural network, Quantum dot neural network, Quantum cell neural network, Quantum bit neural network, and Quantum correlation neural network (Jeswal & Chakraverty, 2019).

In summary, quantum neural networks are an emerging research field that integrates classical neural networks with quantum computing concepts. By leveraging the advantages of quantum computing and the characteristics of quantum superposition, quantum neural networks hold the potential to exhibit unique advantages in handling complex tasks, optimising models, and solving optimisation problems (Gong et al., 2023; Zhou, Li, et al., 2023). Researchers have proposed various types of quantum neural network models, which bring new opportunities and challenges to the fields of quantum computing and machine learning. With advancements in quantum computing technology and further research, our understanding and application of quantum neural networks will continue to expand, opening up new possibilities for future scientific and technological advancements.

Quantum neural network is widely used in fields such as image recognition (Li et al., 2020), superconducting processor (Pan et al., 2023), handwritten digit recognition (Zhou, Liu, et al., 2023) and Medical RNA recognition (Rebentrost et al., 2018). Especially, to better understand the application of quantum neural network, Rebentrost et. al. (Rebentrost et al., 2018) propose a quantum hopfield neural network, and apply as a RNA genetic sequence recogniser.

Figure 6 (Rebentrost et al., 2018) illustrates an RNA sequencing application of the quantum Hopfield neural network. The authors encoded 50 RNA bases from 8 strands of the H1N1 influenza A virus into the matrix W. They ran the quantum Hopfield neural network using partial information from a randomly selected subset of RNA bases from the first strand. The trial involved both the conventional classical method (dotted line) and the matrix-inversion method (solid line) in executing the quantum Hopfield neural network. Over 1000 repetitions and across different levels of partial data, the average Hamming distance to the true data was computed. This demonstrates that the quantum Hopfield neural network can effectively perform a swap test to match the target state with the encoded H1N1 virus.

Figure 6. The quantum Hopfield neural network serves as a content-addressable memory system applied in RNA recognition.

3.3. Quantum k-Nearest Neighbors algorithm

The k-Nearest Neighbors algorithm offers the advantage of being non-parametric, distinguishing itself from many other supervised learning algorithms by not making any assumptions about the data distribution. However, for large datasets, its computational cost can become very high. In the algorithm process, the calculation of distances between instances is the most frequent step, so quantum KNN focuses on how to accelerate the evaluation of distances between two instances. In 2013, Lloyd introduced a supervised learning quantum algorithm based on distance metrics (Hu et al., 2020). This algorithm demonstrates an exponential acceleration over classical algorithms by enhancing the distance computation through quantum methodologies. Since then, researchers have found various methods to parallelly compute the similarity of samples using the principles of quantum superposition and subsequently utilise suitable search algorithms to find the minimum distance. Li et al. proposed in 2022 to first leverage quantum computing for parallel computation of Hamming distance (Li, Lin, et al., 2021). In order to find the minimum distance, they then devised a fundamental sub-algorithm for discovering the minimum value of an unordered integer series. Additionally, Gao et al. introduced the quantum k-nearest neighbours (KNN) algorithm based on the Mahalanobis distance (Gao et al., 2022). In 2023, Feng et al. also introduced the quantum KNN algorithm based on polar distance (Feng et al., 2023), while Zardini et al. proposed the quantum KNN algorithm based on estimated Euclidean distance (Zardini et al., 2023).

The quantum KNN algorithm unfolds in three primary stages: initialising quantum states, similarity computation, and utilising search algorithms to find the k most similar values.

The quantum KNN algorithm applies the principles of quantum computation to classical KNN. Since quantum computation relies on quantum bits, the first step is to set up two quantum states: A single quantum state α is allocated for storing data concerning all instances to be categorised, while a separate quantum state β is designated for storing data corresponding to instances serving as samples. These mathematical formulas for the superposition states α and β are both quantum states (Chen et al., 2015): (22) $\begin{array}{rl}\alpha & =\frac{1}{\sqrt{d}}\sum _{i=1}^d|i〉(\sqrt{1-v_{0i}^2}|0〉+v_{0i}|1〉)|1〉,\end{array}$(22) (23) $\begin{array}{rl}\text{β}& =\frac{1}{\sqrt{M}}\sum _{j=1}^M|j〉\frac{1}{\sqrt{d}}\sum _{i=1}^d|i〉(\sqrt{1-v_{\mathrm{ji}}^2}|0〉+v_{\mathrm{ji}}|1〉).\end{array}$(23) In the equation, d represents the dimension of vector v_j, while v_ji represents the i-th attribute value within vector v_j. In this context, v₀ signifies the vector representing the instance for classification, whereas the vectors denoted as v_j (where j = 1, 2, … , M) represent the known class sample instances.

To prepare such quantum states, we first need to normalise the sample data to obtain the vector v_j. During the preparation process, an Oracle operator calculation $0|j〉|i〉|0〉=|j〉|i〉|v_{\mathrm{ji}}〉$ is required.

The vector data information is encoded within the amplitudes of the quantum states α and β, as described in the preparation process detailed earlier. Through an exchange operation utilising a unitary matrix (unitary gate), the cosine distance between the vectors recorded in the amplitudes of the quantum states is calculated. The quantum circuit for the exchange operation using the unitary gate is shown in Figure 7.

Figure 7. Controlled swap unitary gate.

By applying the controlled swap gate, we obtain the quantum state γ. (24) $\frac{1}{\sqrt{M}}\sum _{j=1}^M|j〉[\sqrt{1-d(v_0,v_j)}|0〉+\sqrt{d(v_0,v_j)}|1〉].$(24) Next, the quantum state undergoes the amplitude estimation algorithm, converting the likeness between each sample data v_j and v₀ into quantum bits. This results in the following quantum state: (25) $\sigma =\frac{1}{\sqrt{M}}\sum _{j=1}^M|j〉||v_j-v_0|〉.$(25)

The process to retrieve the k most similar points to the unlabelled data from the quantum state σ are as follows:

• Let the set K = { K₁, K₂, … , K_k } represent the k points that need to be determined as the most similar to v₀. Begin by selecting k points at random from the set of sample data and allocate them to the collection K.

• Using the Grover algorithm, we can obtain a point v_j from the quantum state σ in such a way that v_j is closer to the unlabelled point v₀ compared to any point v_Kx present in set K: d(v₀, v_j) < d(v₀, v_Kx), where x∈[1, k].

• Replace the point K_x in set K with the j-th point, where x is max{d(v₀, v_Kx)}, and x ∈ [1, k].

• Repetition of steps 2 and 3 will result in the smallest k points, which can be obtained by finding the closest k points by bringing the value of t down until it equals 0.

Quantum k-Nearest Neighbors is widely used in fields such as Hamming code (Li, Lin, et al., 2021), image recognition (Wiebe et al., 2014b; Zhou et al., 2021) and quantum communication network (Hahn et al., 2022). Especially, to better understand the application of quantum k-Nearest Neighbors, Wiebe et. al. (Wiebe et al., 2014a) apply it in the real-world of binary handwritten image recognition, noting that its classification accuracy competes effectively with classical methodologies.

As a concrete example of quantum k-Nearest Neighbors classification, Figure 8 (Wiebe et al., 2014a) shows the training data of handwritten digits expressed with an example of 25 digits, each of which is represented by a 256-dimensional feature vector of pixel values. Figure 9 (Wiebe et al., 2014a) processes raw data into feature vectors and distance for handwriting recognition. The experiment compares the quantum k-Nearest Neighbors algorithm to the classic nearest centroid algorithm, as a function of noise in the distance computation. The accuracy is averaged across classification of 100 random test examples using 2000 training points, and the quantum k-Nearest Neighbors significantly outperforms the classic nearest centroid by roughly 20% in the the low-noise regime.

Figure 8. An example of 25 handwritten digits. Each digit is stored as a 256-pixel greyscale image and represented as a unit vector with N = 256 features.

Figure 9. Schematic process for processing raw data into feature vectors into distances for handwriting recognition.

3.4. Quantum principal component analysis

Machine learning is widely employed across diverse fields, including computer vision and natural language processing and so on, with a particular emphasis on handling datasets characterised by high dimensionality. These high-dimensional datasets are rife with irrelevant information. As the dimensionality of data increases, some algorithms may become ineffective in high-dimensional spaces. Hence, dimensionality reduction is crucial. It simplifies data by eliminating irrelevant details, aiding models in more effective learning (Li, Liang, et al., 2023; Liang, Yang, et al., 2023; Xu et al., 2022). Principal Component Analysis (PCA), a commonly applied method for reducing dimensionality, transforming data through linear combinations to create uncorrelated principal components while retaining most of the essential information. The schematic representation of the Principal Component Analysis (PCA) method is illustrated in Figure 10.

Figure 10. The schematic diagram of Principal Component Analysis (PCA). Note that the image was originally presented in (Fakultat, 2006).

Lloyd et al. introduced the Quantum Principal Component Analysis algorithm in 2013(Lloyd et al., 2013b). A better QPCA technique was put forth in 2019 by Lin et al., which does not suffer from the issue of needing a lot of samples to determine the principal components (Lin et al., 2019). A low-complexity Quantum Principal Component Analysis (QPCA) algorithm was put forth by Chen et al. in 2022. It performs dimensionality reduction similarly to the state-of-the-art QPCA by extracting only the primary components of the data matrix onto a quantum register rather than the entire data matrix. As a result, the required number of measurement samples can be significantly reduced (He et al., 2022). The QPCA algorithm has been proven to offer exponential acceleration compared to any known classical algorithm (Zhang & Ni, 2020). The programme for QPCA is as follows:

Before encoding the classical data as quantum states, we first normalise the classical data. In order to continue, we remove the mean value, designated as $\overline{x}=\frac{1}{M}\sum _{i=1}^M{x}^i$, from a set of N-dimensional vectors $x^i\to x^i-\overline{x}$, where i is a number between 1 and M. Then, we normalise as follows $x^i=\frac{x^i}{|x^i|}$, $|x|=\sqrt{\sum _{k=1}^N{x}_k^2}$, and proceed to encode the classical data as quantum states $x\to |x〉=\sum _{k=1}^N{x}_k|k〉$.

In the QPCA algorithm, the heart of the process lies in phase estimation, necessitating the creation of two input components: the controlled U operator and the quantum state ρ. The density matrix shown below can be used to define the quantum state ρ: (26) $\rho =\frac{1}{M}\sum _{i=1}^M|x^i〉〈x^i|=\frac{1}{M}\left[\begin{array}{cccc}{\displaystyle \sum _{i=1}^M{x}_1^i{x}_1^i}& {\displaystyle \sum _{i=1}^M{x}_1^i{x}_2^i}& \cdots & {\displaystyle \sum _{i=1}^M{x}_1^i{x}_N^i}\\ {\displaystyle \sum _{i=1}^M{x}_2^i{x}_1^i}& {\displaystyle \sum _{i=1}^M{x}_2^i{x}_2^i}& \cdots & {\displaystyle \sum _{i=1}^M{x}_2^i{x}_N^i}\\ ⋮& ⋮& \ddots & ⋮\\ {\displaystyle \sum _{i=1}^M{x}_N^i{x}_1^i}& {\displaystyle \sum _{i=1}^M{x}_N^i{x}_2^i}& \cdots & {\displaystyle \sum _{i=1}^M{x}_N^i{x}_N^i}\end{array}\right].$(26) The controlled U operator can be created by following the procedure described in reference (Lloyd et al., 2013b), resulting in an expression of the form e^-iρt.

Once we possess these inputs, we can proceed with the application of the quantum phase estimation subroutine. In the QPCA algorithm, as opposed to directly implementing the unitary operator U = e^-iρt onto the eigenvectors, we employ the density matrix ρ on the eigenvectors. The ensuing outcome of this quantum phase estimation application is: (27) $\sum _{j=1}^M{\lambda }^j|{\tilde{\lambda }}^j〉〈{\tilde{\lambda }}^j|\otimes |x^j〉〈x^j|.$(27) Finally, we can learn more about the properties of the eigenvectors by sampling from this state.

Quantum principal component analysis algorithm is widely used in fields such as visualisation (Sarmina et al., 2023), data dimension reduction (He et al., 2022), Hamming code (Lloyd et al., 2013b) and data processing (Lin et al., 2019).

He and colleagues have introduced a quantum principal component analysis algorithm with reduced complexity to enhance its practical applicability (He et al., 2022). They conducted simulations of the algorithm on the IBM quantum computing platform, as illustrated in Figure 11, using a 2*2 matrix. The experimental results demonstrate that the proposed quantum principal component analysis algorithm achieves a runtime approximately 3/5 faster than the current state of the art.

Figure 11. Simulation circuit of proposed qPCA for the 2 × 2 matrix A0 on IBM QX.

3.5. Quantum k-Means algorithm

Lloyd and colleagues presented a quantum adaptation of the Nearest Centroid algorithm (Lloyd et al., 2013a). This quantum Nearest Centroid algorithm operates as an essential component within the classical k-Means clustering algorithm, offering a remarkable exponential acceleration over the traditional method. Additionally, Hiroshi Ohno explored the potential of leveraging quantum computation to expedite the k-Means algorithm in handling extensive datasets in 2022 (Ohno, 2022). He proposed a quantum-enhanced k-Means algorithm where centroid computation is eliminated. Poggiali et al. designed a hybrid quantum algorithm for k-Means (Poggiali et al., 2022a; Poggiali et al., 2022b). The main idea is to calculate the distances between records in the input dataset quantumly, improving parallelism and saving time. The Manhattan distance-based quantum k-Means method (QKMM) was proposed by Wu et al. (Wu et al., 2021). The distances between each training vector and each of the k cluster centroids are calculated, and then the nearest cluster centroid is chosen, in the two primary steps of the QKMM method. k-Medoids, the k-Means algorithm, a highly effective clustering technique, is commonly utilised in various domains such as data mining, image processing, and pattern recognition. The cluster assignment and centre updating operations at the heart of the k-Medoids technique can be computationally demanding for large datasets. The Manhattan distance between any two locations can be calculated using a quantum subroutine described by Li et al. in 2023. They then assigned all of the points in parallel to the closest centres, obtaining quicker performance compared to classical methods (Li, Liu, et al., 2023).

Once the starting state is established in the quantum k-Means algorithm, the similarity between the data points and cluster centroids is assessed by determining their Euclidean distance from one another. The Grover optimisation approach is utilised to choose the nearest centroid, and the swap-test programme is frequently used as a subroutine for distance calculation. Subsequent to this, iterations of the cluster centroids are executed until convergence is achieved (Jadhav et al., 2023). In 2022, Kavitha et al. introduced a quantum k-Means clustering method for detecting heart disease using quantum circuit methods and introduced a quantum k-Means algorithm (Kavitha & Kaulgud, 2021). The entire quantum k-Means clustering process includes three subroutines: distance calculation, clustering update, and centroid update. The algorithm proceeds through the following specific steps.

• (1) distance calculation

Before calculating the distance, classical data is encoded into quantum data. In the quantum context, the distribution of data points is determined by measuring the overlap function of the superposition state between two vectors. The conversion of the output of this function into quantum form requires an exchange test. The quantum switch test circuit is shown in Figure 12.

Figure 12. Quantum-swap-test-circuit Where $|\varphi 〉$ represents the centroid cluster, and $|\psi 〉$represents the new data point for which its corresponding cluster is to be identified.

Prepare the states $|\psi 〉$and $|\varphi 〉$.as well as the control qubit |0〉. (28) $\begin{array}{l}|S〉=|0,|\psi 〉,|\varphi 〉〉\\ \begin{array}{cc}& {\to }_H\end{array}\frac{1}{\sqrt{2}}(|0,|\psi 〉,|\varphi 〉〉+|1,|\psi 〉,|\varphi 〉)\\ \begin{array}{cc}& {\to }_{\mathrm{SWAP}}\end{array}\frac{1}{\sqrt{2}}(|0,|\psi 〉,|\varphi 〉〉+|1,|\varphi 〉,|\psi 〉)\\ \begin{array}{cc}& {\to }_H\end{array}\frac{1}{2}|0〉(|\psi ,\varphi 〉+|\varphi ,\psi 〉)+\frac{1}{2}|1〉(|\psi ,\varphi 〉-|\varphi ,\psi 〉).\end{array}$(28) The probability of measuring |0〉 for the ancillary quantum bit after the final measurement is as follows. (29) $P(|0〉)={\left|\frac{1}{2}〈0|0〉(|\psi ,\varphi 〉+|\varphi ,\psi 〉)+\frac{1}{2}〈0|1〉(|\psi ,\varphi 〉-|\varphi ,\psi 〉)\right|}^2=\frac{1}{2}+\frac{1}{2}|〈\psi |\varphi 〉{|}^2.$(29) When measuring and controlling quantum bits, a measurement probability of 1 indicates that the two states are indistinguishable. Conversely, a measurement probability of 0.5 signifies that the two states are either orthogonal or non-overlapping. Similarly, the probability for |1〉 is as follows. (30) $P(|1〉)=\frac{1}{2}-\frac{1}{2}|〈\psi |\varphi 〉{|}^2.$(30) The distance (similarity) between the two states is as follows. (31) $\mathrm{Dist}=d^2{r}_{max}^4\left(2P\right(|0〉-1\left)\right).$(31)

Where, d represents the input non-zero data points, and r_max signifies the maximum boundary of any feature in the data.

• (2) Cluster updation

Following the distance calculation, the new data points are then allocated to their closest cluster using the nearest neighbour function (Sergioli et al., 2017).

• (3) Centroid updation

The updating of centroids is similar to classical k-Means, both determined by the average of all data points.

Repeat the above three steps until the centroids do not change, indicating convergence.

In recent years, researchers have proposed various quantum k-Means algorithms. We'll just introduce one of them here. In general, these algorithms are anticipated to offer quantum advantages and create more effective clustering algorithms that outperform traditional classification algorithms in terms of time complexity as quantum technology develops (Liang, Li, et al., 2023).

The quantum k-Means algorithm has seen extensive application. For example, UlHaq et al. introduced the utilisation of quantum k-Means clustering in analyzing RNA-Seq data from cancer transcriptomes (UlHaq & Bonny, 2020). The entire process involves three primary stages: quantum bit data encoding, distance calculation, and quantum bit measurement. To leverage the quantum algorithm, classical data needs to be translated into quantum bits, establishing the mapping of classical information onto Bloch’s sphere for quantum representation. After the data is normalised, it is then mapped to the Bloch’s sphere as illustrated in Figure 13. Quantum gates and circuits are used to process and evaluate quantum correlations, including operations such as the U3 gate, Hadamard gate, and the controlled-swap gate with three quantum bits. Finally, these operations estimate the distance between two data points directly from the measurement results. The quantum measurement process uses randomisation techniques to decompose the superposed quantum bits into definite states, in order to find the most probable state and make the output of the quantum computer useful.

Figure 13. Data mapping diagram.

The data for this experiment is selected from the “Gene Expression Cancer RNASeq” dataset, which includes gene expression data from different types of cancer patients such as BRCA, KRIC, COAD, LUAD, and PRAD. In the experimental cancer analysis, the quantum k-Means clustering algorithm showcased an average accuracy of 94.8%.

3.6. Comparison and analysis

As summarised in Table 4, we briefly compare the advantages and disadvantages of existing quantum machine learning algorithms of quantum support vector machine, quantum neural network, quantum k-nearest neighbours, quantum principal component analysis and quantum k-Means.

Table 4. Comparison of advantages and disadvantages of existing quantum machine learning algorithms.

QML	Ref	Advantages	Disadvantages
Quantum support vector machine	(Zhang et al., 2022)	Achieving exponential acceleration, improving accuracy, and more effectively handling noisy or incomplete data, resulting in reliable outcomes.	The constraint imposed by the input matrix reduces applicability, and dependence on the choice of hyperparameters leads to sensitivity in parameter selection.
Quantum neural network	(Abel et al., 2022)	Quantum training is accomplished in a single annealing step, reducing training time and enhancing classification performance simultaneously.	The use of a quantum annealer restricts connectivity between quantum bits, making it challenging to handle higher-order polynomials.
Quantum k-Nearest Neighbors	(Li, Lin, et al., 2021)	Efficient, higher precision, and broader applicability of Hamming distance.	Introducing additional parameters increases computational complexity and imposes limitations on quadratic acceleration when dealing with high dimensions.
Quantum principal component analysis	(Li, Chai, et al., 2021)	Exponentially higher efficiency compared to classical methods, requiring fewer auxiliary quantum resources.	Susceptible to decoherence effects in auxiliary quantum bits.
Quantum k-Means	(Ohno, 2022)	Providing exponential acceleration, reducing computational demands, and minimising energy consumption.	Due to hardware limitations, noise, and error rates, the feasibility of the algorithm on quantum hardware is constrained.

Specially, the quantum support vector machine algorithm achieves exponential speedup compared to classical support vector machines, exhibiting high accuracy, robustness to sparse solutions and low complexity, thereby enhancing the efficiency of training and classification tasks. However, it is constrained when the structure of the input matrix is unclear, supports only binary classification, and is sensitive to hyper parameter selection, requiring careful tuning for optimal performance (Zhang et al., 2022).

The quantum neural network algorithm efficiently trains full quantum neural networks by stably finding the global minimum in a single annealing step, ensuring optimal performance. Nevertheless, its applicability is limited by the number and connectivity of qubits, as well as the effectiveness in handling higher-order problems (Abel et al., 2022).

The quantum k-nearest neighbours algorithm achieves significant acceleration through quantum computation, utilising Hamming distance for parallel computation and applicable to a broader range of data types. However, it introduces an additional parameter t, which increases computational complexity, struggles to achieve quadratic speedup with high-dimensional sample vectors, and requires further research for the effective selection of parameter k. (Li, Lin, et al., 2021).

The quantum principal component analysis algorithm demonstrates remarkable acceleration in data dimensionality reduction, especially when the covariance matrix is low-rank and stored in quantum states, showing exponential efficiency improvement. Nevertheless, it is susceptible to the decoherence of auxiliary quantum bits (Li, Chai, et al., 2021).

The quantum k-Means algorithm achieves exponential speedup in large-scale data scenarios through quantum computation while maintaining clustering performance comparable to classical k-Means algorithms. However, the algorithm depends on quantum computation, and the feasibility on actual quantum computers are subject to hardware limitations (Ohno, 2022).

4. Application

Combining quantum computing and machine learning, quantum machine learning is a new topic that may provide solutions to challenging pattern recognition and data processing issues. Quantum machine learning holds significant application potential across various domains, including network security, privacy preservation, bioinformatics, computational biology, image recognition and processing, computational chemistry, natural language processing, and more. Although quantum machine learning has already touched upon multiple interdisciplinary fields, it is still in its nascent stages (Liu et al., 2024).

Exploring quantum machine learning for image recognition stands as a burgeoning frontier within research. Although there is currently no fully quantum-based image recognition method, some studies have indicated that the characteristics of quantum computing can be used to improve the performance of image recognition tasks in classical machine learning applications. In 2023, Wei and colleagues delivered an extensive survey on the application of quantum machine learning techniques in medical image analysis, summarising the advancements made in the past decade (Wei et al., 2023). Experimental data from various sources have consistently demonstrated the powerful capabilities of quantum machine learning in image recognition for medical purposes.

In the fields of biology and bioinformatics, problems such as protein structure prediction, protein binding, gene expression, and drug design are both crucial and highly challenging. Traditional computational methods often struggle to address these issues rapidly and accurately. However, quantum machine learning is poised to make significant breakthroughs in this domain. By employing quantum algorithms for parallel computation, it can swiftly process extensive protein and DNA sequence data, thereby enhancing the accuracy and efficiency of prediction and design. Drug development involves intricate and time-consuming procedures. Quantum machine learning presents a means to expedite this process. It applies to screening and designing drug molecules, predicting their activity, affinity, and potential side effects. Predicting drug reactions is also crucial, and using quantum neural networks to personalise medical treatment plans for each patient and predict drug responses can effectively address the limitations of many traditional medical procedures (Sagingalieva et al., 2022).

Network security is a critical field that is essential for protecting the security of networks and data. Quantum machine learning can be utilised in the field of network security to enhance precision and effectiveness of detecting intrusions, analyzing threat intelligence, and predicting vulnerabilities (Xie et al., 2024). Existing machine learning models can only identify malicious users after data breaches occur, making the estimation of malicious users a primary task. The QM-MUP (Quantum Machine Learning-based Malicious User Prediction) model introduces quantum machine learning techniques to classify user requests, ensuring secure data handling and communication (Gupta et al., 2022). Moreover, quantum cryptography plays a role in network security by ensuring data confidentiality.

4.1. Image analysis

As stated in the introduction, as data continues to grow, the processing capacity of machine learning approaches its limits. Inspired by quantum mechanics, researchers have begun to explore quantum computing models that leverage the parallelism and acceleration properties of quantum systems for processing n-dimensional data. High-dimensional data in medical imaging can also undergo medical analysis through Quantum Machine Learning (Liang, Li, et al., 2024).

Brain tumour is one of the most challenging diseases, and its prevalence in India continues to rise. The gold standard for early brain tumour diagnosis has been limited to targeted biopsy of brain tissue samples. However, this technique suffers from significant unreliability, potentially leading to medical misdiagnoses and, if mishandled, the risk of cerebral hemorrhage or even fatality.

Given the limitations of conventional diagnostic methods, non-invasive imaging modalities such as Magnetic Resonance Imaging (MRI) have garnered substantial attention for diagnosing brain tumours. However, a significant challenge of this technology is that under low-illumination conditions, it is often difficult to visually distinguish the characteristic features of tumour cells from surrounding normal cells. This is because tumour cells exhibit complex and unpredictable patterns: they lack a fixed shape, vary in size, and their location within the tumour is also uncertain. Consequently, in such situations, it becomes challenging to distinctly identify tumour cells. Kanimozhi and colleagues proposed a CNN-quantum hybrid transfer learning approach for brain tumour recognition (Kanimozhi et al., 2022).

In this approach, as MRI images are classical, we need to employ classical methods for data preprocessing and feature extraction, Certainly, the method involves using a pre-trained ResNet-18 CNN to extract essential features from the data. These extracted features are later integrated with a custom-designed variational quantum circuit, which functions as the classifier. The overall structure of the framework is depicted in Figure 14.

Figure 14. Brain tumour image analysis.

The model consists of five stages:

Stage 1: Adjust the image dimensions and normalise pixel intensities.

Stage 2: Utilise the ResNet-18 CNN architecture to extract relevant clinical features from brain MRI images.

Stage 3: In this stage, the ResNet-18 CNN-derived features are adaptively encoded to align with the quantum circuit's qubit specifications. Specifically, the process involves transforming 512 features into 4 quantum bits to suit the quantum system's requirements.

Stage 4: The entire variational quantum circuit is composed of an embedding layer, multiple variational layers, and a final measurement layer. In the embedding layer, a Hadamard gate is used to initialise the quantum bits in a balanced superposition of $|0〉$ and $|1〉$, followed by quantum bit rotations based on input parameters. In the quantum variational layers, a sequence of rotation gates forms the rotation layer, and entanglement layers are composed of CNOT gates to fulfil the training process. Overall, the training process involves data passing through alternating rotation and entanglement gates in the quantum variational layers. In the measurement layer, data is measured using Pauli-Z gates to convert quantum data into classical data for classification.

Stage 5: In this stage, the 4-bit classical output features obtained are transmitted to the final fully connected layer to generate two-dimensional target output class predictions.

Experiments were conducted by repeating training and testing the model with varying quantum depths. The model improved diagnostic accuracy, achieving a maximum classification accuracy of 0.967.

4.2. Drug efficacy prediction

Cancer stands as a substantial global health concern and continues to be a prominent cause of mortality in numerous nations. Chemotherapy remains a viable treatment approach, despite its notable adverse effects on the human body. Different treatment approaches should be tailored to each individual cancer patient based on their specific circumstances (Yang et al., 2023; Zhang et al., 2023). Drug therapy is also a treatment modality, but drugs come with their respective side effects. Therefore, it is highly worthwhile to research and find the optimal drug dosages that maximise effectiveness while minimising side effects.

The evolution of machine learning in drug efficacy prediction is ongoing. However, the accuracy of such predictions relies heavily on extensive medical data, which can be challenging to obtain due to its scarcity in real-world scenarios. The IC50 metric is used to measure drug responses, indicating the half-maximal inhibitory concentration. Determining the IC50 values can assist researchers in evaluating the effectiveness of compounds and provide crucial references for subsequent drug development. Sagingalieva et al. proposed that the Hybrid Quantum Neural Network (HQNN) can perform well in drug response prediction even with small datasets (Sagingalieva et al., 2022).

Sagingalieva and team introduced the HQNN model, comprised of three sub-networks: a cell line representation neural network, a drug representation neural network, and a quantum neural network. The overall framework of the Hybrid Quantum Neural Network is illustrated in Figure 15, as shown below.

Figure 15. The overall framework diagram of the hybrid quantum neural network for drug efficacy prediction.

In the neural network for cell line representation, the cell lines are encoded based on their genomic features using a one-dimensional convolutional layer. The presence of a genomic mutation is denoted by “1,” while “0” signifies normal genetic attributes. Following this one-dimensional encoding, the information goes through three Max Pooling layers and recurrent convolutional layers, culminating in a 128-dimensional vector derived from the fully connected layer.

In the neural network for drug representation, drugs are selected from the PubChem database and represented as graphs, with each node having corresponding chemical features. RDKit software is used to convert drug data into molecular graphs. Since drug representation here is in the form of graphs, a graph convolutional neural network is employed in this subnetwork to learn features. The network takes two matrices as input: the feature matrix X, which contains 78-dimensional binary feature vectors for each atom in the molecule, and the adjacency matrix A, which encodes atom connections. The network is composed of three graph convolutional layers, each succeeded by a ReLU activation function. These layers help in learning interactions between atoms, generating C-dimensional features for each node (atom). Following the last graph convolutional layer, the application of global max-pooling aims to extract important features for the entire molecule. Subsequently, the output is transformed into a 128-dimensional vector through two fully connected layers of 312 and 1024 dimensions. This enables an effective representation of drug molecules.

Within the quantum neural network, the quantum layer comprises an embedding layer, a variational layer, and measurement components. The initial step involves merging cell and drug data into a 256-node feature vector. This 256-dimensional feature vector is encoded in groups of eight features, utilising angle embedding to map the vector into the quantum Hilbert space. Subsequently, it undergoes a variational layer composed of trainable parameters in the form of rotations and controlled gates. Finally, by applying CNOT operations to specific chemical compound/cell line pairs, the z-measurements are propagated to all quantum bits. Multiplicative and additive trainable parameters are employed to enhance the z-measurement values for predicting IC50 values.

Through experimental validation, it was demonstrated that the HQNN outperformed classical models with the same architecture by 15%.

4.3. Cybersecurity

Intrusion detection is a critical task in network security, aiming to timely detect and prevent unauthorised access, malicious activities, or attackers’ intrusion into computer systems, networks, or applications. Applying quantum machine learning to intrusion detection can improve detection accuracy, handle high-dimensional data, monitor and respond in real-time, and develop more secure intrusion detection systems and encryption algorithms when facing quantum attacks (Cai et al., 2023; Liang, Xie, et al., 2024).

Chen and colleagues introduced a novel approach for intrusion detection by combining the quantum-inspired Ant Lion Optimization (ALO) with the k-Means algorithm (Chen et al., 2020). The ALO method, inspired by the hunting strategy of ant lion larvae, is a nature-inspired algorithm. This method transforms the optimisation problem into the process of ants randomly walking in the search space. Based on this, ALO performs optimisation through five main steps: setting traps, trapping ants in the traps, capturing prey, and reconstructing traps. Here, traps refer to a collection of local optimal solutions containing multiple solutions, while capturing prey refers to selecting the global optimal solution from the traps and using it as the starting point for the next search. Through these steps, ALO can effectively search for the global optimum in the search space.

The general idea of the algorithm is:

• Define fundamental algorithmic parameters, including the population size, maximum iteration count, and other relevant factors.

• Initialise the population using a double-chain quantum-inspired encoding scheme. Each ant's position is represented as a sequence of quantum bits, and their positions are initialised using a random walk method.

• Evaluate the fitness of each ant based on a fitness function and select the elite ants and ant lions with higher fitness.

• Use quantum rotation gates to transform the quantum bit sequences and update the ants’ positions. Adjust the positions based on the positions of the elites and ant lions.

• Select ants using a roulette wheel selection method and update their positions based on random walks, including moving towards elites and ant lions, moving towards other ants, or random wandering.

• Evaluate the fitness of each ant based on the fitness function again and select the elite ants and ant lions with higher fitness.

• Continue executing steps 4 through 6 until the predefined limit of iterations is achieved

• Finally, choose either the fittest ant or ant lion as the cluster centre and use the classical k-Means algorithm for clustering.

The QALO-K algorithm, a commonly used anomaly detection approach, was applied to evaluate the KDD Cup 99 dataset for cluster analysis. The algorithm achieved a highest accuracy of 98.69%, a detection rate of 98.58%, and a significantly lower false positive rate compared to the results obtained with k-Means and ALO-K algorithms.

5. Challenge

While quantum machine learning holds promise for significant advancements in data processing, optimisation, and pattern recognition, it also presents several challenges that warrant attention. In this section, we present several noteworthy challenges, including algorithm design, quantum hardware limitations, quantum datasets and encoding, quantum landscape, and quantum noise and decoherence, as shown in Table 5.

Table 5. The challenges of different quantum machine learning algorithms.

Challenges	Ref	Remark
Algorithm design	(Koch et al., 2020) (Cerezo et al., 2022)	Designing quantum machine learning algorithms poses a significant challenge as it involves adapting classical principles creatively, optimising for quantum hardware, and addressing susceptibility to noise and errors.
Quantum hardware limitation	(Houssein et al., 2022) (Abohashima et al., 2020) (Garcia et al., 2022)	Many algorithms currently remain predominantly in the simulation stage, and existing quantum computers still lack in terms of quality and scalability. Limited quantum resources constrain widespread usage, necessitating improvements in quantum hardware to overcome challenges in algorithm execution.
Quantum dataset and encoding	(Lloyd et al., 2020) (Huang et al., 2022) (Wei et al., 2023)	Encoding classical data into quantum states is a crucial engineering challenge. Due to the scarcity of quantum datasets, testing relies on easily preparable quantum data. Testing quantum machine learning with classical datasets may not yield significant quantum advantages, as quantum datasets are more likely to demonstrate quantum advantages over classical datasets.
Quantum landscapes	(Arrasmith et al., 2022) (McClean et al., 2018)	Optimising parameters for global optimum is a key task in quantum machine learning. However, the presence of local minima and barren plateaus, especially in large-scale problems, poses a critical challenge, limiting the trainability of quantum models.
Quantum noise and decoherence	(Li et al., 2020) (Liang et al., 2021) (Bultrini et al., 2023)	Quantum machine learning faces NISQ computer noise challenges, requiring innovative error correction and robust algorithms for reliability. Noise disrupts information propagation, affecting various model aspects, making strategies like reducing hardware errors, partial quantum correction, and resilient model design crucial for addressing noise issues.

5.1. Algorithm design

The advancement of quantum machine learning necessitates the creation of novel algorithms tailored for quantum computations. Many classical machine learning algorithms cannot be directly translated into forms that are suitable for quantum computation due to the non-determinism and quantum parallelism (Wang, Fontana, et al., 2021; Zhang et al., 2018). Designing quantum machine learning algorithms optimised for both quantum hardware and algorithmic characteristics is a crucial and difficult task (Koch et al., 2020). Designing quantum machine learning algorithms suitable for quantum computers involves a thorough understanding of quantum hardware characteristics, along with the creative adaptation of classical machine learning principles into forms compatible with quantum computation. This process requires rethinking the mathematical expressions of problems, data encoding, and the overall algorithmic structure to fully harness the benefits of quantum computing, such as non-determinism and quantum parallelism (Cerezo et al., 2022). Simultaneously, quantum machine learning algorithms must take into account the susceptibility of quantum systems to noise and errors, necessitating the design of noise-robust data encoding and error correction mechanisms (Wang, Fang, et al., 2021; Wang, Jin, et al., 2021; Shen et al., 2019).

5.2. Quantum hardware limitations

In recent years, substantial theoretical advancements have been made in quantum machine learning algorithms, accompanied by experimentation on quantum simulators. Nevertheless, most of these algorithms have remained at the simulation stage and have not been tested in actual experiments to determine their performance compared to classical methods (Simoes et al., 2023). The current lack of sufficient quality, speed, and scalability in existing quantum computers hinders them from fully harnessing their potential. Figure 16 illustrates the number of quantum bits (qubits) implemented by various technology companies such as Rigetti, IBM, Q Wave, Xanadu, Google, and Microsoft (Houssein et al., 2022). This chart illustrates significant advancements in recent years concerning the number of quantum bits (qubits). However, current research is still largely limited to simulation and small-scale computation. The scarcity of quantum computing resources has hindered their widespread availability and support from experimental hardware, posing challenges for broad usage and deployment (Abohashima et al., 2020). Therefore, improvements in quantum hardware are necessary (Garcia et al., 2022). Many proposed algorithms have not been successfully executed in a quantum environment. Despite theoretical evidence of significant performance improvements of quantum algorithms over traditional ones, it remains challenging to analyze the performance difference between quantum machine learning and classical machine learning in actual experiments. Despite significant advancements in quantum technology, achieving universal fault-tolerant quantum computers with a high qubit count remains a challenging objective. The precise quantity of logical qubits required for a quantum computer to outperform extremely powerful classical computers in terms of computational capabilities is currently uncertain.

Figure 16. Progress of Quantum Computers and the Quantity of Quantum Bits.

The creation of interface tools that enable the quantum encoding of classical information, like quantum random access memory (QRAM), is a critical issue for quantum machine learning. QRAM is a key technology in quantum computing that enables efficient data storage and retrieval. However, due to the absence of a direct quantum counterpart to traditional classical Random Access Memory, the practical implementation of QRAM has been an ongoing research area. QRAM is crucial for efficient computations on large-scale quantum systems. While methods have been proposed to encode classical information into the reduction of quantum probability amplitudes, further efforts are needed in the physical implementation (Arunachalam et al., 2015). Researchers have analyzed and improved QRAM from various perspectives, yet an effective QRAM method capable of storing arbitrary quantum states has not been achieved.

5.3. Quantum dataset and encoding

Encoding classical data into quantum states is a prerequisite for quantum machine learning. This involves how to represent and encode data, as well as how to transmit and process data in a quantum system (Cai et al., 2023; Diao et al., 2023; Long et al., 2023). In quantum computing, data encoding typically refers to the transformation of classical information into probability amplitudes that describe quantum states. Probability amplitude is a mathematical construct used in quantum mechanics to detail the probability relationship between system states and different states. However, encoding data into probability amplitudes can lead to the creation of a huge Hilbert space, thereby increasing the complexity of extracting classical information from quantum information. Numerous contemporary quantum machine learning methodologies continue to depend on classical dataset testing through the integration of classical data into quantum states. Nevertheless, these embedding schemes must adhere to specific ideal properties, such as inner products that pose challenges for simulation on classical computers and ensuring distinguishable states within Hilbert space. In current embedding approaches, while one property of data embedding may be fulfilled, it could fall short in satisfying another (Lloyd et al., 2020). Therefore, encoding classical data into quantum data is a key engineering challenge, especially crucial for achieving efficient quantum machine learning (Wei et al., 2023). Moreover, handling data access and preprocessing in quantum machine learning requires careful consideration of quantum characteristics, presenting a unique challenge. Consequently, quantum machine learning encounters intricacies in encoding and processing data, navigating specific demands stemming from quantum properties. Additionally, testing quantum machine learning with classical datasets may not yield significant quantum advantages, as quantum datasets are more likely to demonstrate quantum advantages over classical datasets (Huang et al., 2022). However, quantum datasets are currently limited, so Quantum Machine Learning (QML) relies on easily preparable quantum datasets for data analysis and testing purposes.

5.4. Quantum landscapes

The optimisation of parameters in quantum machine learning is a critical task, revolving around the pursuit of global optima. Quantum landscape theory is designed to grasp and dissect optimisation problems and loss function landscapes within the realm of quantum machine learning (Arrasmith et al., 2022). This theoretical framework delves into aspects such as local minima and barren plateaus—phenomena that hinder the trainability of Quantum Machine Learning (QML).

In the realm of quantum machine learning, as depicted in Figure 17(a), the loss function may manifest multiple local minima. Within optimisation algorithms, being confined to a local minimum can lead to the solution obtained not being the global optimum. Although quantum algorithms often exhibit advantages in tackling problems that pose challenges for classical computation, the existence of local minima presents analogous optimisation challenges for quantum algorithms.

Figure 17. Quantum landscapes.

Barren plateaus describe a phenomenon in which the quantum loss function landscape flattens exponentially with respect to the problem size (McClean et al., 2018), as depicted in Figure 17(b). In such instances, the gradient of the loss function diminishes significantly, giving rise to the vanishing gradient problem throughout the training process, thereby intensifying the difficulty of optimisation. The existence of barren plateaus implies that training quantum models may necessitate exponential resources (e.g. computational or measurement resources), posing impractical challenges for real-world quantum computers. This limitation constrains the applicability of quantum machine learning algorithms, especially in the context of large-scale problems.

5.5. Quantum noise and decoherence

Quantum computers currently in use fall under the category known as Noisy Intermediate-Scale Quantum (NISQ) computers. These devices, characterised by a higher level of noise and a moderate count of quantum bits (qubits), encounter challenges in maintaining the stability of quantum states during practical operations. This susceptibility to noise, errors, and decoherence is outlined by Li et al. . In quantum systems, noise and decoherence are unavoidable sources of errors (Li et al., 2020). Noise can originate from environmental interactions and physical defects in quantum bits, while decoherence refers to the loss of quantum coherence over time, leading to the degradation of quantum states. To ensure the reliability of quantum computation, error correction techniques and robust algorithms are required to mitigate these challenges. Particularly, in the case of limited qubits, addressing noise becomes an urgent issue (Liang et al., 2021).

The presence of noise disrupts the propagation of information in quantum computers, particularly noticeable in circuits with extended runtime and greater depth. This disruption has repercussions on various aspects of models leveraging quantum computers, encompassing data set preparation and the circuits utilised for computing quantum kernels. Approaches to mitigate issues stemming from noise in quantum machine learning encompass efforts to decrease hardware error rates, employ partial quantum error correction (Bultrini et al., 2023), opt for relatively shallow quantum circuits (Wang, Fontana, et al., 2021), and embrace error mitigation techniques. Additionally, another strategy involves crafting Quantum Machine Learning (QML) models with characteristics resilient to noise, ensuring that noise does not substantially alter the position of the model's minimum.

6. Conclusion and future work

This paper offers a glimpse into recent advancements in quantum machine learning research, it is crucial to acknowledge that this is not a comprehensive review. Before delving into the intersection of machine learning and quantum information processing, it is important to have a deep understanding of both fields. So far, most of the research in quantum machine learning has come from experts in classical machine learning and quantum information processing, and this interdisciplinary collaboration has been very encouraging and successful. Quantum machine learning, as a cross-disciplinary research area combining principles of quantum mechanics and machine learning, holds important research significance and value in the following aspects: Firstly, it leverages the high parallelism of quantum computing to enhance the ability of machine learning in handling, analyzing, and mining large-scale data. Secondly, by drawing inspiration from quantum mechanics principles, it promotes the innovation and development of new machine learning algorithms. Thirdly, it proposes new research approaches by borrowing from traditional machine learning algorithms, thus driving further advancements in the field of quantum mechanics.

In our future work, firstly, we will enhance the performance and reliability of quantum machine learning, aiming to play an important role in solving complex problems, optimising algorithms and achieving more practical applications and breakthroughs. Despite being in its early stages, researchers have made progress in addressing issues such as noise and errors in quantum computers, data accessibility, and algorithm design when it comes to quantum machine learning. Secondly, we will utilise the parallelism of quantum computing and introduce quantum k-Means for feature matching of key points to achieve image copy and paste detection, thereby reducing algorithm complexity and improving robustness. Because in the process of copy and paste detection in the same image, the extracted image features have a large number of key points and multiple feature dimensions, which leads to high computational complexity, high feature matching time consumption, and poor robustness of classical machine learning algorithms.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Key Research and Development Program of China [grant numbers 2022YFA1602200 and 2021YFA1000600], the National Natural Science Foundation of China [grant number 62072170 and 62202156], the international partnership program of the Chinese Academy of Sciences [grant number 211134KYSB20200057], the Open Research Fund of Hunan Provincial Key Laboratory of Network Investigational Technology [grant 2020WLZC004], the Science and Technology Project of Hunan Provincial Department of Transportation [grant number 202101], the Key Technologies Research and Development Program of Hunan Province [grant number 2022GK2015], the Natural Science Foundation of Fujian Province [grant number 2023J011460].