Quantum Machine Learning: Algorithms and Complexities

Abstract:

This article provides a comprehensive overview of QML algorithms and explores their complexities. It explores the characteristics of quantum data, hybrid quantum-classical models, variational quantum algorithms, quantum-enhanced reinforcement learning, and the difficulties associated with quantum machine learning. Overall, this article provides a valuable resource for researchers and practitioners interested in understanding the algorithms, complexities, and potential of Quantum Machine Learning, shedding light on its current state and future prospects.

Introduction:

Quantum machine learning, also known as QML, is a blooming field of modern artificial intelligence that integrates quantum computing with machine learning. It aims to enhance traditional machine learning algorithms and develop novel computational methods.

This article examines the inner workings of quantum machine learning and related topics. It includes the fundamentals of quantum computing, quantum machine learning algorithms, the characteristics of quantum data, hybrid quantum-classical models, variational quantum algorithms, quantum-enhanced reinforcement learning, and the difficulties associated with quantum machine learning.

“The fusion of quantum computing and artificial intelligence, paving the way for groundbreaking innovation and endless opportunities.” – Sri Amit Ray

The fusion of quantum computing and artificial intelligence

Quantum machine learning (QML) Algorithms

Quantum machine learning (QML) is a rapidly evolving field that explores the intersection of quantum computing and machine learning. Several algorithms and approaches have been proposed to leverage quantum resources for various machine learning tasks. The prominent quantum machine learning algorithms include:

  1. Quantum Support Vector Machine (QSVM): QSVM is a quantum version of the classical support vector machine algorithm. It aims to classify data points by mapping them into a higher-dimensional feature space using a quantum kernel [4].
  2. Quantum k-Means Clustering: Quantum k-means is a quantum algorithm for clustering data points. It utilizes quantum resources to accelerate the k-means clustering process, which partitions data into k clusters [6].
  3. Quantum Principal Component Analysis (QPCA): QPCA is a quantum variant of the classical principal component analysis algorithm. It leverages quantum computing to extract the principal components and reduce the dimensionality of high-dimensional data [14].
  4. Quantum Generative Adversarial Networks (QGANs): QGANs are quantum versions of generative adversarial networks. They leverage quantum resources to generate synthetic data samples that mimic the distribution of a given dataset.
  5. Quantum Boltzmann Machines (QBMs): QBMs are quantum counterparts of classical Boltzmann machines used for unsupervised learning tasks. They utilize quantum annealing or quantum-inspired optimization to train and model the probability distribution of input data [23].
  6. Variational Quantum Eigensolver (VQE): VQE is a quantum algorithm used for solving optimization problems, including those encountered in machine learning. It employs variational methods and quantum circuits to find the lowest eigenvalue of a given Hamiltonian.
  7. Quantum Neural Networks (QNNs): QNNs are quantum versions of classical neural networks. They leverage the principles of quantum computing to perform computations and training on quantum states. Examples include quantum versions of feedforward and variational quantum neural networks.
  8. Quantum Decision Trees: Quantum decision trees are quantum-based alternatives to classical decision tree algorithms. They aim to efficiently handle large datasets and leverage quantum parallelism for improved decision-making.
  9. Quantum Restricted Boltzmann Machines (QRBMs): QRBMs are quantum counterparts of classical restricted Boltzmann machines. They are used for generative modeling and feature extraction tasks, utilizing quantum resources to model and learn the underlying probability distributions.
  10. Quantum Reinforcement Learning: Quantum reinforcement learning algorithms combine classical reinforcement learning with quantum resources to solve problems such as control, optimization, and sequential decision-making in quantum environments [15].

Nature of Quantum Data

Quantum data possesses distinct characteristics that differentiate it from classical data. Understanding the nature of quantum data is crucial for harnessing its full potential in machine learning applications.

Quantum data representation involves mapping classical data onto quantum states. This process aims to capture richer information and extract meaningful features that may be challenging for classical approaches. By leveraging the power of quantum states, quantum data representation allows for the exploration of novel patterns and insights.

Quantum data preprocessing presents challenges due to noise and errors inherent in quantum systems. Quantum error correction techniques play a crucial role in mitigating these challenges, ensuring the reliability and accuracy of quantum computations. Additionally, quantum data normalization and feature extraction methods are essential to prepare quantum data for subsequent machine learning tasks.

Quantum Computing and QML

Quantum computing serves as the foundation for quantum machine learning. It employs the principles of quantum mechanics to perform computations using quantum bits, or qubits [54]. Understanding the fundamentals of quantum computing is essential to comprehend the potential of quantum machine learning.

Quantum bits, or qubits, are the building blocks of quantum computing. Unlike classical bits that represent information as either 0 or 1, qubits can exist in a superposition of states, simultaneously representing both 0 and 1. This property allows quantum algorithms to process and analyze information in parallel, providing the potential for exponential speedup compared to classical computations [52].

Quantum gates and circuits manipulate qubits to perform quantum computations. These gates, such as the Hadamard gate and the CNOT gate, enable the transformation and entanglement of qubits, facilitating complex calculations. Quantum circuits are sequences of quantum gates that define the flow of quantum operations.

Quantum algorithms designed for machine learning tasks demonstrate the potential of quantum computing. Examples include quantum support vector machines and quantum neural networks. These algorithms leverage the unique properties of qubits, such as superposition and entanglement, to provide computational advantages for tasks like classification, clustering, and regression.

Types of qubits

In quantum computing, qubits are the fundamental units of information that can exist in superposition states, enabling the processing of quantum information. Several types of qubits have been proposed and implemented, each utilizing different physical systems [63]. Here are some common types of qubits:

  1. Superconducting Qubits: Superconducting qubits are based on circuits made of superconducting materials, such as niobium. They operate at extremely low temperatures, close to absolute zero, to maintain superconductivity. Examples include the transmon qubit, flux qubit, and phase qubit. Superconducting qubits are widely used in various quantum computing platforms.
  2. Trapped Ion Qubits: Trapped ion qubits utilize individual ions confined in electromagnetic traps. The internal energy levels of ions act as qubit states, and their manipulation is achieved using laser pulses. Ions such as trapped calcium, magnesium, or ytterbium are commonly used. Trapped ion qubits have achieved remarkable control and coherence times, making them a promising qubit implementation.
  3. Topological Qubits: Topological qubits are a type of qubit that relies on the topological properties of certain materials, such as topological insulators or Majorana fermions. These qubits are robust against certain types of noise and errors due to their inherent protection provided by topology. Topological qubits are still under active research and development.
  4. Quantum Dot Qubits: Quantum dot qubits are based on semiconductor nanostructures that trap electrons in quantum dots. The spin states of the trapped electrons serve as qubits, and their manipulation is achieved through electrostatic gates and magnetic fields. Quantum dot qubits have the advantage of compatibility with existing semiconductor technologies, making them attractive for integration with classical electronic circuits.
  5. Photon Qubits: Photon qubits are qubits encoded in the states of photons, which are particles of light. Photons can be manipulated using various techniques, such as quantum optics and linear optics elements like beam splitters and wave plates. Photon qubits are well-suited for long-distance communication and quantum information processing using optical systems.
  6. Nuclear Magnetic Resonance (NMR) Qubits: NMR qubits utilize the nuclear spins of molecules as qubits. These qubits are manipulated and measured using radiofrequency pulses and NMR techniques. NMR qubits have been extensively used for quantum simulation and small-scale quantum computing experiments.
  7. Diamond Nitrogen-Vacancy (NV) Centers: NV centers in diamond are defects in the crystal lattice where a nitrogen atom replaces a carbon atom, and an adjacent carbon vacancy exists. The electronic and nuclear spin states of the NV centers serve as qubits. These qubits can be controlled and read out using magnetic fields and laser pulses. NV centers are attractive due to their long coherence times and potential for applications in sensing and quantum information processing.

Hybrid Quantum-Classical Models

Hybrid quantum-classical models combine the strengths of both quantum computing and classical machine learning techniques. These models provide a promising approach for solving complex problems that require a balance between quantum processing power and classical control.

Variational quantum algorithms form a prominent class of hybrid models. They employ a combination of classical optimization and quantum computations to solve optimization problems. The Quantum Approximate Optimization Algorithm (QAOA) is an example of a variational quantum algorithm widely used in machine learning tasks.

Quantum-classical neural networks integrate quantum elements into classical neural network architectures. These networks leverage the power of quantum processing for tasks such as image recognition, generative modeling, and pattern recognition. The synergy between quantum and classical components in these networks offers the potential for improved performance and novel applications.

Challenges of Quantum Computing

While quantum machine learning holds great promise, several challenges must be addressed to realize its full potential.

Scalability and qubit limitations are significant challenges in quantum computing. Current quantum hardware faces constraints in terms of the number of qubits and error rates. Developing scalable quantum architectures and improving qubit stability and coherence are ongoing research efforts.

Noise, decoherence, and error correction are critical factors in quantum computing. Quantum systems are susceptible to noise and decoherence, which can lead to errors in computations. Implementing error correction techniques is crucial to maintain the accuracy and reliability of quantum computations.

Algorithm design and optimization are essential for quantum machine learning. Developing efficient quantum algorithms for machine learning problems and optimizing quantum circuits and gate sequences pose significant challenges. Bridging the gap between theoretical algorithms and practical implementations is an active area of research.

Variational quantum algorithms (VQAs)

Variational quantum algorithms (VQAs) are a class of algorithms that leverage the power of variational quantum circuits to solve computational problems. VQAs combine the principles of variational optimization and quantum computing to find solutions to optimization, machine learning, and simulation tasks.

VQAs is a hybrid quantum-classical method in which a quantum processor prepares quantum states and a classical computer performs measurement and optimisation. VQAs are regarded as the optimal algorithm for NISQ because they are noise-tolerant in comparison to other algorithms and provide quantum superiority with only a few hundred qubits.

Researchers [87][89] have investigated circuit-based algorithms for solving optimization problems and determining the ground state energy of complex systems, which were challenging to solve or required a significant amount of time to compute on a conventional computer.

Variational quantum circuits (VQCs)

The key idea behind VQCs is to parameterize the quantum circuit using adjustable parameters, often referred to as variational parameters. These parameters serve as the variables that are optimized during the training process. By adjusting these parameters, the circuit can explore different quantum states and generate a variety of outputs.

The structure of a VQC typically consists of layers of quantum gates, each followed by measurements or additional quantum operations. The gates in the circuit act on specific qubits, often with entangling operations that create quantum correlations among the qubits.

Variational quantum circuits provide a flexible and adaptable framework for combining quantum computations with classical optimization methods. They offer a promising approach to solving complex problems in quantum machine learning, optimization, and quantum chemistry simulations. Ongoing research aims to improve the robustness, scalability, and performance of VQCs, bringing us closer to unlocking their full potential in practical quantum applications.

Quantum Enhanced Reinforcement Learning (QERL)

Quantum-enhanced reinforcement learning (QERL) is an exciting field that combines the principles of reinforcement learning (RL) with the potential advantages offered by quantum computing. QERL aims to leverage the unique properties of quantum systems, such as superposition and entanglement, to enhance the efficiency and effectiveness of RL algorithms.

During the process of quantum-enhanced reinforcement learning, a quantum agent interacts with a classical or quantum environment and occasionally obtains rewards for its activities. This enables the agent to modify its behavior; in other words, it is able to learn what to do in order to gain additional rewards. In some circumstances, either as a result of the agent’s capacity for quantum processing or as a consequence of the possibility of investigating the surroundings in superpositions. 

Conclusion

The subject of machine learning has reached a new threshold with the development of quantum machine learning, which utilizes the power of quantum computing to solve difficult issues and investigate novel approaches to data representation and computation. The issues that are involved with quantum computing may not be going away too soon; nevertheless, continued research and breakthroughs in quantum technologies provide hope that these challenges can be overcome. 

While challenges associated with quantum computing remain, ongoing research and advancements in quantum technologies provide hope for overcoming these obstacles. The potential for quantum machine learning to revolutionize various aspects of artificial intelligence is immense, paving the way for exciting possibilities in the future.

This article provides a comprehensive overview of the fundamental concepts, algorithms, and challenges in quantum machine learning, paving the way for further research and applications in this rapidly evolving field.

References:

  1. Grover, L.K. Quantum Mechanics Helps in Searching for a Needle in a Haystack. Phys. Rev. Lett. 1997, 79, 325–328.
  2. Shor, P.W. Algorithms for Quantum Computation: Discrete Logarithms and Factoring. In Proceedings of the 35th Annual Symposium on Foundation of Computer Science, Washington, DC, USA, 20–22 November 1994; pp. 124–134.
  3. Menneer, T.; Narayanan, A. Quantum-inspired neural networks. In Proceedings of the Neural Information Processing Systems 95, Denver, CO, USA, 27–30 November 1995.
  4. Rebentrost, P.; Mohseni, M.; Lloyd, S. Quantum Support Vector Machine for Big Data Classification. Phys. Rev. Lett. 2014, 113, 130503.
  5. Harrow, A.W.; Hassidim, A.; Lloyd, S. Quantum Algorithm for Linear Systems of Equations. Phys. Rev. Lett. 2009, 103, 150502.
  6. Wiebe, N.; Kapoor, A.; Svore, K. Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning. arXiv 2014, arXiv:1401.2142.
  7. Dang, Y.; Jiang, N.; Hu, H.; Ji, Z.; Zhang, W. Image classification based on quantum K-Nearest-Neighbor algorithm. Quantum Inf. Process. 2018, 17, 1–18.
  8. Schuld, M.; Sinayskiy, I.; Petruccione, F. Prediction by linear regression on a quantum computer. Phys. Rev. A 2016, 94, 022342.
  9. Lu, S.; Braunstein, S.L. Quantum decision tree classifier. Quantum Inf. Process. 2014, 13, 757–770.
  10. Lloyd, S.; Mohseni, M.; Rebentrost, P. Quantum algorithms for supervised and unsupervised machine learning. arXiv 2013, arXiv:1307.0411.
  11. Lloyd, S. Least squares quantization in PCM. IEEE Trans. Inf. Theory 1982, 28, 129–137.
  12. Kerenidis, I.; Landman, J.; Luongo, A.; Prakash, A. q-means: A quantum algorithm for unsupervised machine learning. arXiv 2018, arXiv:1812.03584.
  13. Aïmeur, E.; Brassard, G.; Gambs, S. Quantum speed-up for unsupervised learning. Mach. Learn. 2013, 90, 261–287.
  14. Lloyd, S.; Mohseni, M.; Rebentrost, P. Quantum principal component analysis. Nat. Phys. 2014, 10, 631–633.[Green Version]
  15. Dong, D.; Chen, C.; Li, H.; Tarn, T.-J. Quantum Reinforcement Learning. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 2008, 38, 1207–1220.
  16. Ronagh, P. Quantum algorithms for solving dynamic programming problems. arXiv 2019, arXiv:1906.02229.
  17. McKiernan, K.A.; Davis, E.; Alam, M.S.; Rigetti, C. Automated quantum programming via reinforcement learning for combinatorial optimization. arXiv 2019, arXiv:1908.08054.
  18. Lloyd, S.; Weedbrook, C. Quantum Generative Adversarial Learning. Phys. Rev. Lett. 2018, 121, 040502.
  19. Dallaire-Demers, P.-L.; Killoran, N. Quantum generative adversarial networks. Phys. Rev. A 2018, 98, 012324.
  20. Situ, H.; He, Z.; Wang, Y.; Li, L.; Zheng, S. Quantum generative adversarial network for generating discrete distribution. Inf. Sci. 2020, 538, 193–208.
  21. Huang, H.-L.; Du, Y.; Gong, M.; Zhao, Y.; Wu, Y.; Wang, C.; Li, S.; Liang, F.; Lin, J.; Xu, Y.; et al. Experimental Quantum Generative Adversarial Networks for Image Generation. Phys. Rev. Appl. 2021, 16, 024051.
  22. Chakrabarti, S.; Yiming, H.; Li, T.; Feizi, S.; Wu, X. Quantum Wasserstein generative adversarial networks. arXiv 2019, arXiv:1911.00111.
  23. Kieferová, M.; Wiebe, N. Tomography and generative training with quantum Boltzmann machines. Phys. Rev. A 2017, 96, 062327.
  24. Amin, M.H.; Andriyash, E.; Rolfe, J.; Kulchytskyy, B.; Melko, R. Quantum Boltzmann Machine. Phys. Rev. X 2018, 8, 021050.
  25. Romero, J.; Olson, J.P.; Aspuru-Guzik, A. Quantum autoencoders for efficient compression of quantum data. Quantum Sci. Technol. 2017, 2, 045001.
  26. Khoshaman, A.; Vinci, W.; Denis, B.; Andriyash, E.; Sadeghi, H.; Amin, M.H. Quantum variational autoencoder. Quantum Sci. Technol. 2018, 4, 014001.
  27. Cong, I.; Choi, S.; Lukin, M.D. Quantum convolutional neural networks. Nat. Phys. 2019, 15, 1273–1278.[Green Version]
  28. Kerenidis, I.; Landman, J.; Prakash, A. Quantum algorithms for deep convolutional neural networks. arXiv 2019, arXiv:1911.01117.
  29. Henderson, M.; Shakya, S.; Pradhan, S.; Cook, T. Quanvolutional neural networks: Powering image recognition with quantum circuits. Quantum Mach. Intell. 2020, 2, 2.
  30. Jiang, Z.; Rieffel, E.G.; Wang, Z. Near-optimal quantum circuit for Grover’s unstructured search using a transverse field. Phys. Rev. A 2017, 95, 062317.
  31. Farhi, E.; Goldstone, J.; Gutmann, S. A quantum approximate optimization algorithm. arXiv 2014, arXiv:1411.4028.
  32. Kerenidis, I.; Prakash, A. Quantum gradient descent for linear systems and least squares. Phys. Rev. A 2020, 101, 022316.
  33. Schumacher, B. Quantum coding. Phys. Rev. A 1995, 51, 2738.
  34. Barenco, A.; Bennett, C.H.; Cleve, R.; DiVincenzo, D.P.; Margolus, N.; Shor, P.; Sleator, T.; Smolin, J.A.; Weinfurter, H. Elementary gates for quantum computation. Phys. Rev. A 1995, 52, 3457–3467.
  35. Toffoli, T. Reversible computing. In International Colloquium on Automata, Languages, and Programming; Springer: Berlin/Heidelberg, Germany, 1980; pp. 632–644.
  36. Fredkin, E.; Toffoli, T. Conservative logic. Int. J. Theor. Phys. 1982, 21, 219–253.
  37. Kockum, A.K. Quantum Optics with Artificial Atoms; Chalmers University of Technology: Gothenburg, Sweden, 2014.
  38. Nielsen, M.A.; Chuang, I.; Grover, L.K. Quantum Computation and Quantum Information. Am. J. Phys. 2002, 70, 558–559.
  39. Deutsch, D.; Jozsa, R. Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 1992, 439, 553–558.
  40. Coppersmith, D. An approximate Fourier transform useful in quantum factoring. arXiv 2002, arXiv:quant-ph/0201067.
  41. Xu, N.; Zhu, J.; Lu, D.; Zhou, X.; Peng, X.; Du, J. Quantum Factorization of 143 on a Dipolar-Coupling Nuclear Magnetic Resonance System. Phys. Rev. Lett. 2012, 108, 130501.
  42. Dridi, R.; Alghassi, H. Prime factorization using quantum annealing and computational algebraic geometry. Sci. Rep. 2017, 7, 1–10.
  43. Crane, L. Quantum computer sets new record for finding prime number factors. New Scientist, 13 December 2019.
  44. David, H. “RSA in a “Pre-Post-Quantum” Computing World”. Available online: https://www.f5.com/labs/articles/threat-intelligence/rsa-in-a-pre-post-quantum-computing-world (accessed on 18 May 2023).
  45. Aggarwal, D.; Brennen, G.K.; Lee, T.; Santha, M.; Tomamichel, M. Quantum attacks on Bitcoin, and how to protect against them. arXiv 2017, arXiv:1710.10377. https://arxiv.org/pdf/1710.10377.pdf
  46. Peruzzo, A.; McClean, J.; Shadbolt, P.; Yung, M.-H.; Zhou, X.-Q.; Love, P.J.; Aspuru-Guzik, A.; O’Brien, J.L. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 2014, 5, 4213.
  47. Quantum Machine Learning. Last Modified 6 June 2021. 2021. Available online: https://en.wikipedia.org/wiki/Quantum_machine_learning (accessed on 18 May 2023).
  48. Stoudenmire, E.; Schwab, D.J. Supervised learning with quantum-inspired tensor networks. arXiv 2016, arXiv:1605.05775.
  49. Aaronson, S. The learnability of quantum states. Proc. R. Soc. A Math. Phys. Eng. Sci. 2007, 463, 3089–3114.
  50. Sasaki, M.; Carlini, A. Quantum learning and universal quantum matching machine. Phys. Rev. A 2002, 66, 022303.
  51. Bisio, A.; Chiribella, G.; D’Ariano, G.M.; Facchini, S.; Perinotti, P. Optimal quantum learning of a unitary transformation. Phys. Rev. A 2010, 81, 032324.
  52. Ray, Amit. “The 10 M of Quantum Machine Learning – Amit Ray.” Amit Ray, amitray.com, 12 June 2019, https://amitray.com/the-10-ms-of-quantum-machine-learning/.
  53. Ray, Amit. “Five Key Benefits of Quantum Machine Learning – Amit Ray.” Amit Ray, amitray.com, 29 Apr. 2019, https://amitray.com/five-key-benefits-of-quantum-machine-learning/.
  54. Ray, Amit. “Roadmap for 1000 Qubits Fault-Tolerant Quantum Computers – Amit Ray.” Amit Ray, amitray.com, 15 Mar. 2019, https://amitray.com/roadmap-for-1000-qubits-fault-tolerant-quantum-computers/.
  55. Rossi M, Huber M, Bruß D, Macchiavello C (2013) Quantum hypergraph states. New Journal of Physics 15(11):113022, https://doi.org/10.1088/1367-2630/15/11/113022
  56. da Silva, A.J.; Ludermir, T.B.; de Oliveira, W.R. Quantum perceptron over a field and neural network architecture selection in a quantum computer. Neural Netw. 2016, 76, 55–64.
  57. Tiwari, P.; Melucci, M. Towards a Quantum-Inspired Binary Classifier. IEEE Access 2019, 7, 42354–42372.
  58. Sergioli, G.; Giuntini, R.; Freytes, H. A new quantum approach to binary classification. PLoS ONE 2019, 14, e0216224.
  59. Ding, C.; Bao, T.Y.; Huang, H.L. Quantum-inspired support vector machine. IEEE Trans. Neural Netw. Learn. Syst. 2021, 33, 7210–7222.
  60. Sergioli, G.; Russo, G.; Santucci, E.; Stefano, A.; Torrisi, S.E.; Palmucci, S.; Vancheri, C.; Giuntini, R. Quantum-inspired minimum distance classification in a biomedical context. Int. J. Quantum Inf. 2018, 16, 18400117.
  61. Chen, H.; Gao, Y.; Zhang, J. Quantum k-nearest neighbor algorithm. Dongnan Daxue Xuebao 2015, 45, 647–651.
  62. Yu, C.H.; Gao, F.; Wen, Q.Y. An improved quantum algorithm for ridge regression. IEEE Trans. Knowl. Data Eng. 2019, 33, 858–866.[Green Version]
  63. Ray, Amit. “7 Core Qubit Technologies for Quantum Computing – Amit Ray.” Amit Ray, amitray.com, 10 Dec. 2018, https://amitray.com/7-core-qubit-technologies-for-quantum-computing/.
  64. Ray, Amit. “Quantum Computing with Many Worlds Interpretation Model.” Amit Ray, amitray.com, 30 Jan. 2019, https://amitray.com/quantum-computing-with-many-world-interpretation-scopes-and-challenges/.
  65. Ray, Amit. “Quantum Computer with Superconductivity at Room Temperature.” Amit Ray, amitray.com, 25 Aug. 2018, https://amitray.com/quantum-computing-with-superconductivity-at-room-temperature/.
  66. Sagheer, A.; Zidan, M.; Abdelsamea, M.M. A Novel Autonomous Perceptron Model for Pattern Classification Applications. Entropy 2019, 21, 763.
  67. Adhikary, S.; Dangwal, S.; Bhowmik, D. Supervised learning with a quantum classifier using multi-level systems. Quantum Inf. Process. 2020, 19, 89.
  68. Havlíček, V.; Córcoles, A.D.; Temme, K.; Harrow, A.W.; Kandala, A.; Chow, J.M.; Gambetta, J.M. Supervised learning with quantum-enhanced feature spaces. Nature 2019, 567, 209–212.
  69. Ruan, Y.; Xue, X.; Liu, H.; Tan, J.; Li, X. Quantum Algorithm for K-Nearest Neighbors Classification Based on the Metric of Hamming Distance. Int. J. Theor. Phys. 2017, 56, 3496–3507.
  70. Zhang, D.B.; Zhu, S.L.; Wang, Z.D. Nonlinear regression based on a hybrid quantum computer. arXiv 2018, arXiv:1808.09607.
  71. Cortese, J.A.; Braje, T.M. Loading classical data into a quantum computer. arXiv 2018, arXiv:1803.01958.
  72. Schuld, M.; Killoran, N. Quantum Machine Learning in Feature Hilbert Spaces. Phys. Rev. Lett. 2019, 122, 040504.
  73. Schuld, M.; Petruccione, F. Supervised Learning with Quantum Computers; Springer: Berlin/Heidelberg, Germany, 2018; Volume 17.
  74. Amplitude Amplification. Last Modified 5 April 2021. 2021. Available online: https://en.wikipedia.org/wiki/Amplitude_amplification (accessed on 18 May 2023).
  75. Cerezo, M.; Arrasmith, A.; Babbush, R.; Benjamin, S.C.; Endo, S.; Fujii, K.; McClean, J.R.; Mitarai, K.; Yuan, X.; Cincio, L.; et al. Variational quantum algorithms. Nat. Rev. Phys. 2021, 3, 625–644.
  76. Biamonte, J.; Wittek, P.; Pancotti, N.; Rebentrost, P.; Wiebe, N.; Lloyd, S. Quantum machine learning. Nature 2017, 549, 195–202.
  77. Coyle, B.; Mills, D.; Danos, V.; Kashefi, E. The Born supremacy: Quantum advantage and training of an Ising Born machine. NPJ Quantum Inf. 2020, 6, 60.
  78. Biamonte, J. Universal variational quantum computation. Phys. Rev. A 2021, 103, L030401.
  79. Ray, Amit. “Quantum Computing with Many World Interpretation Model.” Amit Ray, amitray.com, 30 Jan. 2019, https://amitray.com/quantum-computing-with-many-world-interpretation-scopes-and-challenges/.
  80. Ray, Amit. “Quantum Computing with Many World Interpretation Model.” Amit Ray, amitray.com, 30 Jan. 2019, https://amitray.com/quantum-computing-with-many-world-interpretation-scopes-and-challenges/.
  81. Ray, Amit. “Quantum Computing Algorithms for Artificial Intelligence.” Amit Ray, amitray.com, 22 Aug. 2018, https://amitray.com/quantum-computing-algorithms-for-artificial-intelligence/.
  82. Abbas, A.; Sutter, D.; Zoufal, C.; Lucchi, A.; Figalli, A.; Woerner, S. The power of quantum neural networks. Nat. Comput. Sci. 2021, 1, 403–409.
  83. Mitarai, K.; Negoro, M.; Kitagawa, M.; Fujii, K. Quantum circuit learning. Phys. Rev. A 2018, 98, 032309.
  84. Bishwas, A.K.; Mani, A.; Palade, V. An all-pair quantum SVM approach for big data multiclass classification. Quantum Inf. Process. 2018, 17, 282.
  85. Wiebe, N.; Braun, D.; Lloyd, S. Quantum Algorithm for Data Fitting. Phys. Rev. Lett. 2012, 109, 050505.
  86. Kapoor, A.; Wiebe, N.; Svore, K. Quantum perceptron models. arXiv 2016, arXiv:1602.04799.
  87. Schuld, M.; Bocharov, A.; Svore, K.M.; Wiebe, N. Circuit-centric quantum classifiers. Phys. Rev. A 2020, 101, 032308.
  88. Chen, S.Y.-C.; Yang, C.-H.H.; Qi, J.; Chen, P.-Y.; Ma, X.; Goan, H.-S. Variational Quantum Circuits for Deep Reinforcement Learning. IEEE Access 2020, 8, 141007–141024.
  89. Cerezo, M., Arrasmith, A., Babbush, R. et al. Variational quantum algorithms. Nat Rev Phys 3, 625–644 (2021). https://doi.org/10.1038/s42254-021-00348-9
  90. Anguita, D.; Ridella, S.; Rivieccio, F.; Zunino, R. Quantum optimization for training support vector machines. Neural Netw. 2003, 16, 763–770.
  91. Bergholm, V.; Izaac, J.; Schuld, M.; Gogolin, C.; Alam, M.S.; Ahmed, S.; Arrazola, J.M.; Blank, C.; Delgado, A.; Jahangiri, S.; et al. Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv 2018, arXiv:1811.04968.
  92. Aleksandrowicz, G.; Alexander, T.; Barkoutsos, P.; Bello, L.; Ben-Haim, Y.; Bucher, D.; Cabrera-Hernández, F.J.; Carballo-Franquis, J.; Chen, A.; Chen, C.F.; et al. Qiskit: An Open-Source Framework for Quantum Computing. 2019. Available online: https://zenodo.org/record/2562111 (accessed on 18 May 2023).
  93. Buitinck, L.; Louppe, G.; Blondel, M.; Pedregosa, F.; Mueller, A.; Grisel, O.; Niculae, V.; Prettenhofer, P.; Gramfort, A.; Grobler, J.; et al. API design for machine learning software: Experiences from the scikit-learn project. arXiv 2013, arXiv:1309.0238.
Abstract: | Introduction: | Quantum machine learning (QML) Algorithms | Nature of Quantum Data | Quantum Computing and QML | Types of qubits | Hybrid Quantum-Classical Models | Challenges of Quantum Computing | Variational quantum algorithms (VQAs) | Variational quantum circuits (VQCs) | Quantum Enhanced Reinforcement Learning (QERL) | Conclusion | References: |
Tags: