Design and analysis of quantum machine learning: a survey

Figures & data

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.

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)

Figure 1. The overall idea of this paper.

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.

Figure 2. HHL Algorithm Circuit Diagram.

Figure 3. Schematic diagram of Deutsch-Jozsa circuit.

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

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

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

Figure 7. Controlled swap unitary gate.

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.

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

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

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.

Figure 13. Data mapping diagram.

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.

Figure 14. Brain tumour image analysis.

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

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.

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

Figure 17. Quantum landscapes.

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