Data re-uploading for a universal quantum classifier

Adrián Pérez-Salinas; Alba Cervera-Lierta; Elies Gil-Fuster; José I. Latorre

doi:10.22331/q-2020-02-06-226

Data re-uploading for a universal quantum classifier

Adrián Pérez-Salinas^1,2, Alba Cervera-Lierta^1,2, Elies Gil-Fuster³, and José I. Latorre^1,2,4,5

¹Barcelona Supercomputing Center
²Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain
³Dept. Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain.
⁴Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands.
⁵Center for Quantum Technologies, National University of Singapore, Singapore.

Published:	2020-02-06, volume 4, page 226
Eprint:	arXiv:1907.02085v2
Doi:	https://doi.org/10.22331/q-2020-02-06-226
Citation:	Quantum 4, 226 (2020).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Updated version: The authors have uploaded version v3 of this work to the arXiv which may contain updates or corrections not contained in the published version v2. The authors left the following comment on the arXiv:

19 pages, 9 figures

Abstract

A single qubit provides sufficient computational capabilities to construct a universal quantum classifier when assisted with a classical subroutine. This fact may be surprising since a single qubit only offers a simple superposition of two states and single-qubit gates only make a rotation in the Bloch sphere. The key ingredient to circumvent these limitations is to allow for multiple $\textit{data re-uploading}$. A quantum circuit can then be organized as a series of data re-uploading and single-qubit processing units. Furthermore, both data re-uploading and measurements can accommodate multiple dimensions in the input and several categories in the output, to conform to a universal quantum classifier. The extension of this idea to several qubits enhances the efficiency of the strategy as entanglement expands the superpositions carried along with the classification. Extensive benchmarking on different examples of the single- and multi-qubit quantum classifier validates its ability to describe and classify complex data.

Featured image: Results of a classification problem with increasing depth, i. e.: numbers of layers. Results improve as the circuit gets deeper.

Popular summary

In this paper, we show how to use the computational power of a single qubit to solve non-trivial classification problems. We propose a hybrid classical-quantum algorithm based on re-uploading classical data into the angles of the single-qubit unitary gates multiple times along the circuit. Together with the data points, other parameters are introduced into the circuit and adjusted by classically minimizing a cost function. To construct this cost function, we train the circuit to distribute the data points into different regions of the Bloch sphere, one for each class. A particular division of the Bloch sphere accompanies this strategy for maximizing distinguishability between classes.
This procedure cannot provide any quantum advantage as a single qubit can be simulated classically. However, the capability of handling one qubit might be useful as a small piece of larger circuits. Besides, an extension of the algorithm for more qubits and entanglement is also presented in this work. The multi-qubit role remains unexplored and might be a candidate for quantum advantage. A first step analyzed, there exists a trade-off between the number of qubits needed and the times of data re-uploading for classifying, namely layers.
This algorithm is to be compared with a neural network with one hidden layer. Neural Networks re-upload classical data several times, once per hidden neuron, achieving the same kind of processing as in our quantum classifier. Success rates are also comparable for both models.

► BibTeX data

@article{PerezSalinas2020datareuploading,
  doi = {10.22331/q-2020-02-06-226},
  url = {https://doi.org/10.22331/q-2020-02-06-226},
  title = {Data re-uploading for a universal quantum classifier},
  author = {P{\'{e}}rez-Salinas, Adri{\'{a}}n and Cervera-Lierta, Alba and Gil-Fuster, Elies and Latorre, Jos{\'{e}} I.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {4},
  pages = {226},
  month = feb,
  year = {2020}
}

► References

[1] M. Schuld, I. Sinayskiy, and F. Petruccione, Quantum Information Processing 13, 2567 (2014).
https://doi.org/10.1007/s11128-014-0809-8

[2] K. H. Wan, O. Dahlsten, H. Kristjánsson, R. Gardner, and M. S. Kim, npj Quantum Information 3, 36 (2017).
https://doi.org/10.1038/s41534-017-0032-4

[3] E. Torrontegui and J. J. García-Ripoll, EPL (Europhysics Letters) 125, 30004 (2019).
https://doi.org/10.1209/0295-5075/125/30004

[4] N. Wiebe, D. Braun, and S. Lloyd, Physics Review Letters 109, 050505 (2012).
https://doi.org/10.1103/PhysRevLett.109.050505

[5] P. Rebentrost, M. Mohseni, and S. Lloyd, Physics Review Letters 113, 130503 (2014).
https://doi.org/10.1103/PhysRevLett.113.130503

[6] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature 549, 195 (2017).
https://doi.org/10.1038/nature23474

[7] E. Farhi and H. Neven, ``Classification with quantum neural networks on near term processors,'' (2018), arXiv:1802.06002 [quant-ph].
arXiv:1802.06002

[8] M. Schuld, A. Bocharov, K. Svore, and N. Wiebe, ``Circuit-centric quantum classifiers,'' (2018), arXiv:1804.00633 [quant-ph].
arXiv:1804.00633

[9] V. Havlíček, A. D. Córcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, Nature 567, 209 (2019).
https://doi.org/10.1038/s41586-019-0980-2

[10] M. Schuld and N. Killoran, Physics Review Letters 122, 040504 (2019).
https://doi.org/10.1103/PhysRevLett.122.040504

[11] V. Giovannetti, S. Lloyd, and L. Maccone, Physics Review Letters 100, 160501 (2008).
https://doi.org/10.1103/PhysRevLett.100.160501

[12] K. Hornik, Neural Networks 4, 251 (1991).
https://doi.org/10.1016/0893-6080(91)90009-t

[13] R. Ghobadi, J. S. Oberoi, and E. Zahedinejhad, ``The power of one qubit in machine learning,'' (2019), arXiv:1905.01390 [quant-ph].
arXiv:1905.01390

[14] J. Gil Vidal and D. Oliver Theis, ``Input redundancy for parameterized quantum circuits,'' (2019), arXiv:1901.11434 [quant-ph].
arXiv:1901.11434

[15] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Physics Review A 98, 032309 (2018).
https://doi.org/10.1103/PhysRevA.98.032309

[16] C. W. Helstrom, Quantum detection and estimation theory / Carl W. Helstrom (Academic Press New York, 1976) pp. ix, p. : 309.
https://nla.gov.au/nla.cat-vn617918

[17] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 10th ed. (Cambridge University Press, New York, NY, USA, 2011).
https://doi.org/10.1017/CBO9780511976667

[18] G. Cybenko, Mathematics of Control, Signals, and Systems 2, 303 (1989).
https://doi.org/10.1007/bf02551274

[19] B. C. Hall, Lie Groups, Lie Algebras, and Representations An Elementary Introduction (Graduate Texts in Mathematics, 222 (2nd ed.), Springer, 2015).
https://doi.org/10.1007/978-3-319-13467-3

[20] M. A. Nielsen, Neural networks and deep learning, Vol. 25 (Determination press USA, 2015).
http://neuralnetworksanddeeplearning.com/

[21] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, SIAM Journal on Scientific Computing 16, 1190 (1995).
https://doi.org/10.1137/0916069

[22] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay, Journal of Machine Learning Research 12, 2825 (2011).
https://www.scipy.org/

[23] E. Jones, T. Oliphant, P. Peterson, et al., ``SciPy: Open source scientific tools for Python,'' https://www.scipy.org/ (2001).
https://www.scipy.org/

[24] A. Pérez-Salinas, ``Quantum classifier with data re-uploading,'' https://github.com/AdrianPerezSalinas/universal_qlassifier (2019).
https://github.com/AdrianPerezSalinas/universal_qlassifier

[25] S. Ahmed, ``Data-reuploading classifer,'' https://pennylane.ai/qml/app/tutorial_data_reuploading_classifier.html (2019).
https://pennylane.ai/qml/app/tutorial_data_reuploading_classifier.html

[26] J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Quantum Science and Technology 4, 014008 (2018).
https://doi.org/10.1088/2058-9565/aad3e4

Cited by

[1] Mohammad Pirhooshyaran and Tamás Terlaky, "Quantum circuit design search", Quantum Machine Intelligence 3 2, 25 (2021).

[2] Parfait Atchade-Adelomou and Guillermo Alonso-Linaje, "Quantum-enhanced filter: QFilter", Soft Computing 26 15, 7167 (2022).

[3] Hiroshi Yano, Yudai Suzuki, Rudy Raymond, and Naoki Yamamoto, 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 11 (2020) ISBN:978-1-7281-8969-7.

[4] Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planagumà, and José I. Latorre, "Quantum unary approach to option pricing", Physical Review A 103 3, 032414 (2021).

[5] Gerhard Hellstem, 2021 IEEE 18th International Conference on Software Architecture Companion (ICSA-C) 1 (2021) ISBN:978-1-6654-3910-7.

[6] Saurabh Kumar, Siddharth Dangwal, Soumik Adhikary, and Debanjan Bhowmik, 2021 International Joint Conference on Neural Networks (IJCNN) 1 (2021) ISBN:978-1-6654-3900-8.

[7] Ryan LaRose and Brian Coyle, "Robust data encodings for quantum classifiers", Physical Review A 102 3, 032420 (2020).

[8] Tarun Dutta, Adrián Pérez-Salinas, Jasper Phua Sing Cheng, José Ignacio Latorre, and Manas Mukherjee, "Single-qubit universal classifier implemented on an ion-trap quantum device", Physical Review A 106 1, 012411 (2022).

[9] Vasilis Belis, Samuel González-Castillo, Christina Reissel, Sofia Vallecorsa, Elías F. Combarro, Günther Dissertori, Florentin Reiter, C. Biscarat, S. Campana, B. Hegner, S. Roiser, C.I. Rovelli, and G.A. Stewart, "Higgs analysis with quantum classifiers", EPJ Web of Conferences 251, 03070 (2021).

[10] Marco Pistoia, Syed Farhan Ahmad, Akshay Ajagekar, Alexander Buts, Shouvanik Chakrabarti, Dylan Herman, Shaohan Hu, Andrew Jena, Pierre Minssen, Pradeep Niroula, Arthur Rattew, Yue Sun, and Romina Yalovetzky, 2021 IEEE/ACM International Conference On Computer Aided Design (ICCAD) 1 (2021) ISBN:978-1-6654-4507-8.

[11] Aikaterini Gratsea and Patrick Huembeli, "Exploring quantum perceptron and quantum neural network structures with a teacher-student scheme", Quantum Machine Intelligence 4 1, 2 (2022).

[12] Md Habib Ullah, Rozhin Eskandarpour, Honghao Zheng, and Amin Khodaei, "Quantum computing for smart grid applications", IET Generation, Transmission & Distribution gtd2.12602 (2022).

[13] Long Hin Li, Dan-Bo Zhang, and Z.D. Wang, "Quantum kernels with Gaussian state encoding for machine learning", Physics Letters A 436, 128088 (2022).

[14] Maria Schuld and Nathan Killoran, "Is Quantum Advantage the Right Goal for Quantum Machine Learning?", PRX Quantum 3 3, 030101 (2022).

[15] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum algorithms", Reviews of Modern Physics 94 1, 015004 (2022).

[16] Marcello Benedetti, Brian Coyle, Mattia Fiorentini, Michael Lubasch, and Matthias Rosenkranz, "Variational Inference with a Quantum Computer", Physical Review Applied 16 4, 044057 (2021).

[17] Pere Mujal, Johannes Nokkala, Rodrigo Martínez-Peña, Gian Luca Giorgi, Miguel C Soriano, and Roberta Zambrini, "Analytical evidence of nonlinearity in qubits and continuous-variable quantum reservoir computing", Journal of Physics: Complexity 2 4, 045008 (2021).

[18] M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, "Variational quantum algorithms", Nature Reviews Physics 3 9, 625 (2021).

[19] Teppei Suzuki and Michio Katouda, "Predicting toxicity by quantum machine learning", Journal of Physics Communications 4 12, 125012 (2020).

[20] Thomas Hubregtsen, Josef Pichlmeier, Patrick Stecher, and Koen Bertels, "Evaluation of parameterized quantum circuits: on the relation between classification accuracy, expressibility, and entangling capability", Quantum Machine Intelligence 3 1, 9 (2021).

[21] Mahabubul Alam and Swaroop Ghosh, "QNet: A Scalable and Noise-Resilient Quantum Neural Network Architecture for Noisy Intermediate-Scale Quantum Computers", Frontiers in Physics 9, 755139 (2022).

[22] William Cappelletti, Rebecca Erbanni, and Joaquin Keller, 2020 IEEE International Conference on Quantum Computing and Engineering (QCE) 22 (2020) ISBN:978-1-7281-8969-7.

[23] Kushal Batra, Kimberley M. Zorn, Daniel H. Foil, Eni Minerali, Victor O. Gawriljuk, Thomas R. Lane, and Sean Ekins, "Quantum Machine Learning Algorithms for Drug Discovery Applications", Journal of Chemical Information and Modeling 61 6, 2641 (2021).

[24] Beng Yee Gan, Daniel Leykam, and Dimitris G. Angelakis, "Fock state-enhanced expressivity of quantum machine learning models", EPJ Quantum Technology 9 1, 16 (2022).

[25] Fabio Valerio Massoli, Lucia Vadicamo, Giuseppe Amato, and Fabrizio Falchi, "A Leap among Quantum Computing and Quantum Neural Networks: A Survey", ACM Computing Surveys 3529756 (2022).

[26] Carlos Bravo-Prieto, "Quantum autoencoders with enhanced data encoding", Machine Learning: Science and Technology 2 3, 035028 (2021).

[27] Javier Villalba-Diez, Joaquín Ordieres-Meré, Ana González-Marcos, and Aintzane Soto Larzabal, "Quantum Deep Learning for Steel Industry Computer Vision Quality Control.", IFAC-PapersOnLine 55 2, 337 (2022).

[28] Stefano Mangini, Alessia Marruzzo, Marco Piantanida, Dario Gerace, Daniele Bajoni, and Chiara Macchiavello, "Quantum neural network autoencoder and classifier applied to an industrial case study", Quantum Machine Intelligence 4 2, 13 (2022).

[29] Takahiro Goto, Quoc Hoan Tran, and Kohei Nakajima, "Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces", Physical Review Letters 127 9, 090506 (2021).

[30] Abhinav Anand, Matthias Degroote, and Alán Aspuru-Guzik, "Natural evolutionary strategies for variational quantum computation", Machine Learning: Science and Technology 2 4, 045012 (2021).

[31] Leonardo Banchi, Jason Pereira, and Stefano Pirandola, "Generalization in Quantum Machine Learning: A Quantum Information Standpoint", PRX Quantum 2 4, 040321 (2021).

[32] Mahabubul Alam, Satwik Kundu, Rasit Onur Topaloglu, and Swaroop Ghosh, 2021 IEEE/ACM International Conference On Computer Aided Design (ICCAD) 1 (2021) ISBN:978-1-6654-4507-8.

[33] Kathleen E. Hamilton, Emily Lynn, and Raphael C. Pooser, " Mode connectivity in the loss landscape of parameterized quantum circuits", Quantum Machine Intelligence 4 1, 10 (2022).

[34] Oriel Kiss, Francesco Tacchino, Sofia Vallecorsa, and Ivano Tavernelli, "Quantum neural networks force fields generation", Machine Learning: Science and Technology 3 3, 035004 (2022).

[35] Adrián Pérez-Salinas, David López-Núñez, Artur García-Sáez, P. Forn-Díaz, and José I. Latorre, "One qubit as a universal approximant", Physical Review A 104 1, 012405 (2021).

[36] Toshiaki Koike-Akino, Pu Wang, and Ye Wang, 2022 IEEE 12th Sensor Array and Multichannel Signal Processing Workshop (SAM) 360 (2022) ISBN:978-1-6654-0633-8.

[37] Matthias C. Caro, Hsin-Yuan Huang, M. Cerezo, Kunal Sharma, Andrew Sornborger, Lukasz Cincio, and Patrick J. Coles, "Generalization in quantum machine learning from few training data", Nature Communications 13 1, 4919 (2022).

[38] Eraraya Ricardo Muten, Togan Tlimakhov Yusuf, and Andrei Voicu Tomut, 2021 IEEE International Conference on Quantum Computing and Engineering (QCE) 82 (2021) ISBN:978-1-6654-1691-7.

[39] N. Schetakis, D. Aghamalyan, P. Griffin, and M. Boguslavsky, "Review of some existing QML frameworks and novel hybrid classical–quantum neural networks realising binary classification for the noisy datasets", Scientific Reports 12 1, 11927 (2022).

[40] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer, "Effect of data encoding on the expressive power of variational quantum-machine-learning models", Physical Review A 103 3, 032430 (2021).

[41] Paolo Braccia, Filippo Caruso, and Leonardo Banchi, "How to enhance quantum generative adversarial learning of noisy information", New Journal of Physics 23 5, 053024 (2021).

[42] Jakob S. Kottmann, Abhinav Anand, and Alán Aspuru-Guzik, "A feasible approach for automatically differentiable unitary coupled-cluster on quantum computers", Chemical Science 12 10, 3497 (2021).

[43] Gerhard Hellstern, "Analysis of a hybrid quantum network for classification tasks", IET Quantum Communication 2 4, 153 (2021).

[44] Parfait Atchade Adelomou, Daniel Casado Fauli, Elisabet Golobardes Ribé, and Xavier Vilasís-Cardona, "Quantum case-based reasoning (qCBR)", Artificial Intelligence Review (2022).

[45] S. Mangini, F. Tacchino, D. Gerace, D. Bajoni, and C. Macchiavello, "Quantum computing models for artificial neural networks", Europhysics Letters 134 1, 10002 (2021).

[46] Alba Cervera-Lierta, Jakob S. Kottmann, and Alán Aspuru-Guzik, "Meta-Variational Quantum Eigensolver: Learning Energy Profiles of Parameterized Hamiltonians for Quantum Simulation", PRX Quantum 2 2, 020329 (2021).

[47] Adrián Pérez-Salinas, Juan Cruz-Martinez, Abdulla A. Alhajri, and Stefano Carrazza, "Determining the proton content with a quantum computer", Physical Review D 103 3, 034027 (2021).

[48] Matthias C. Caro, Elies Gil-Fuster, Johannes Jakob Meyer, Jens Eisert, and Ryan Sweke, "Encoding-dependent generalization bounds for parametrized quantum circuits", Quantum 5, 582 (2021).

[49] Maja Franz, Lucas Wolf, Maniraman Periyasamy, Christian Ufrecht, Daniel D. Scherer, Axel Plinge, Christopher Mutschler, and Wolfgang Mauerer, "Uncovering Instabilities in Variational-Quantum Deep Q-Networks", Journal of the Franklin Institute (2022).

[50] Liangliang Fan and Haozhen Situ, "Compact data encoding for data re-uploading quantum classifier", Quantum Information Processing 21 3, 87 (2022).

[51] Stavros Efthymiou, Sergi Ramos-Calderer, Carlos Bravo-Prieto, Adrián Pérez-Salinas, Diego García-Martín, Artur Garcia-Saez, José Ignacio Latorre, and Stefano Carrazza, " Qibo: a framework for quantum simulation with hardware acceleration", Quantum Science and Technology 7 1, 015018 (2022).

[52] Manas Sajjan, Junxu Li, Raja Selvarajan, Shree Hari Sureshbabu, Sumit Suresh Kale, Rishabh Gupta, Vinit Singh, and Sabre Kais, "Quantum machine learning for chemistry and physics", Chemical Society Reviews 51 15, 6475 (2022).

[53] Soumik Adhikary, "Entanglement assisted training algorithm for supervised quantum classifiers", Quantum Information Processing 20 8, 254 (2021).

[54] Carlos Bravo-Prieto, Josep Lumbreras-Zarapico, Luca Tagliacozzo, and José I. Latorre, "Scaling of variational quantum circuit depth for condensed matter systems", Quantum 4, 272 (2020).

[55] Abhinav Anand, Jonathan Romero, Matthias Degroote, and Alán Aspuru‐Guzik, "Noise Robustness and Experimental Demonstration of a Quantum Generative Adversarial Network for Continuous Distributions", Advanced Quantum Technologies 4 5, 2000069 (2021).

[56] Maria Schuld and Francesco Petruccione, Quantum Science and Technology 217 (2021) ISBN:978-3-030-83097-7.

[57] Korbinian Kottmann, Friederike Metz, Joana Fraxanet, and Niccolò Baldelli, "Variational quantum anomaly detection: Unsupervised mapping of phase diagrams on a physical quantum computer", Physical Review Research 3 4, 043184 (2021).

[58] Beng Yee Gan, Daniel Leykam, and Dimitris G. Angelakis, Conference on Lasers and Electro-Optics JW1A.73 (2021) ISBN:978-1-943580-91-0.

[59] Mihai-Bebe Simion, Dan Selisteanu, and Dorin Sendrescu, 2022 23rd International Carpathian Control Conference (ICCC) 319 (2022) ISBN:978-1-6654-6636-3.

[60] Li Ding and Lee Spector, Proceedings of the Genetic and Evolutionary Computation Conference Companion 2190 (2022) ISBN:9781450392686.

[61] Francisco Orts, Gloria Ortega, and Ester M. Garzón, Lecture Notes in Computer Science 13353, 122 (2022) ISBN:978-3-031-08759-2.

[62] Suguru Endo, Zhenyu Cai, Simon C. Benjamin, and Xiao Yuan, "Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation", Journal of the Physical Society of Japan 90 3, 032001 (2021).

[63] Patrick Huembeli and Alexandre Dauphin, "Characterizing the loss landscape of variational quantum circuits", Quantum Science and Technology 6 2, 025011 (2021).

[64] Teresa Sancho-Lorente, Juan Román-Roche, and David Zueco, "Quantum kernels to learn the phases of quantum matter", Physical Review A 105 4, 042432 (2022).

[65] Nhat A. Nghiem, Samuel Yen-Chi Chen, and Tzu-Chieh Wei, "Unified framework for quantum classification", Physical Review Research 3 3, 033056 (2021).

[66] Carlos Bravo-Prieto, Julien Baglio, Marco Cè, Anthony Francis, Dorota M. Grabowska, and Stefano Carrazza, "Style-based quantum generative adversarial networks for Monte Carlo events", Quantum 6, 777 (2022).

[67] Gabriele Agliardi and Enrico Prati, "Optimal Tuning of Quantum Generative Adversarial Networks for Multivariate Distribution Loading", Quantum Reports 4 1, 75 (2022).

[68] Francisco Orts, Gloria Ortega, Ernestas Filatovas, and Ester M. Garzón, "Implementation of three efficient 4-digit fault-tolerant quantum carry lookahead adders", The Journal of Supercomputing 78 11, 13323 (2022).

[69] Toshiaki Koike-Akino, Pu Wang, and Ye Wang, ICC 2022 - IEEE International Conference on Communications 654 (2022) ISBN:978-1-5386-8347-7.

[70] Hiroshi Yano, Yudai Suzuki, Kohei Itoh, Rudy Raymond, and Naoki Yamamoto, "Efficient Discrete Feature Encoding for Variational Quantum Classifier", IEEE Transactions on Quantum Engineering 2, 1 (2021).

[71] Andrea Skolik, Sofiene Jerbi, and Vedran Dunjko, "Quantum agents in the Gym: a variational quantum algorithm for deep Q-learning", Quantum 6, 720 (2022).

[72] Philip Easom-Mccaldin, Ahmed Bouridane, Ammar Belatreche, and Richard Jiang, "On Depth, Robustness and Performance Using the Data Re-Uploading Single-Qubit Classifier", IEEE Access 9, 65127 (2021).

[73] Bryan Liu, Toshiaki Koike-Akino, Ye Wang, and Kieran Parsons, ICC 2022 - IEEE International Conference on Communications 1775 (2022) ISBN:978-1-5386-8347-7.

[74] Maria Schuld and Francesco Petruccione, Quantum Science and Technology 147 (2021) ISBN:978-3-030-83097-7.

[75] Evan Peters, João Caldeira, Alan Ho, Stefan Leichenauer, Masoud Mohseni, Hartmut Neven, Panagiotis Spentzouris, Doug Strain, and Gabriel N. Perdue, "Machine learning of high dimensional data on a noisy quantum processor", npj Quantum Information 7 1, 161 (2021).

[76] Parfait Atchade-Adelomou, Guillermo Alonso-Linaje, Jordi Albo-Canals, and Daniel Casado-Fauli, "qRobot: A Quantum Computing Approach in Mobile Robot Order Picking and Batching Problem Solver Optimization", Algorithms 14 7, 194 (2021).

[77] P A M Casares and M A Martin-Delgado, "A quantum active learning algorithm for sampling against adversarial attacks", New Journal of Physics 22 7, 073026 (2020).

[78] Maria Schuld and Francesco Petruccione, Quantum Science and Technology 177 (2021) ISBN:978-3-030-83097-7.

[79] Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran, "Quantum embeddings for machine learning", arXiv:2001.03622, (2020).

[80] Tobias Haug, Chris N. Self, and M. S. Kim, "Quantum machine learning of large datasets using randomized measurements", Machine Learning: Science and Technology 4 1, 015005 (2023).

[81] Noah L. Wach, Manuel S. Rudolph, Fred Jendrzejewski, and Sebastian Schmitt, "Data re-uploading with a single qudit", arXiv:2302.13932, (2023).

[82] Atchade Parfait Adelomou, Elisabet Golobardes Ribe, and Xavier Vilasis Cardona, "Using the Parameterized Quantum Circuit combined with Variational-Quantum-Eigensolver (VQE) to create an Intelligent social workers' schedule problem solver", arXiv:2010.05863, (2020).

[83] Smit Chaudhary, Patrick Huembeli, Ian MacCormack, Taylor L. Patti, Jean Kossaifi, and Alexey Galda, "Towards a scalable discrete quantum generative adversarial neural network", arXiv:2209.13993, (2022).

[84] Junde Li, Mahabubul Alam, Congzhou M Sha, Jian Wang, Nikolay V. Dokholyan, and Swaroop Ghosh, "Drug Discovery Approaches using Quantum Machine Learning", arXiv:2104.00746, (2021).

[85] S. Shin, Y. S. Teo, and H. Jeong, "Exponential data encoding for quantum supervised learning", Physical Review A 107 1, 012422 (2023).

[86] S. Carrazza, S. Efthymiou, M. Lazzarin, and A. Pasquale, "An open-source modular framework for quantum computing", Journal of Physics Conference Series 2438 1, 012148 (2023).

[87] Francisco Orts, Gloria Ortega, Elías F. Combarro, Ignacio F. Rúa, and Ester M. Garzón, "Optimized quantum leading zero detector circuits", Quantum Information Processing 22 1, 28 (2023).

[88] Carlos A. Riofrío, Oliver Mitevski, Caitlin Jones, Florian Krellner, Aleksandar Vučković, Joseph Doetsch, Johannes Klepsch, Thomas Ehmer, and Andre Luckow, "A performance characterization of quantum generative models", arXiv:2301.09363, (2023).

[89] Bálint Máté, Bertrand Le Saux, and Maxwell Henderson, "Beyond Ansätze: Learning Quantum Circuits as Unitary Operators", arXiv:2203.00601, (2022).

[90] Muhammad Kashif and Saif Al-Kuwari, "The impact of cost function globality and locality in hybrid quantum neural networks on NISQ devices", Machine Learning: Science and Technology 4 1, 015004 (2023).

[91] William Cappelletti, Rebecca Erbanni, and Joaquín Keller, "Polyadic Quantum Classifier", arXiv:2007.14044, (2020).

[92] Yoshiaki Kawase, Kosuke Mitarai, and Keisuke Fujii, "Parametric t-stochastic neighbor embedding with quantum neural network", Physical Review Research 4 4, 043199 (2022).

[93] Anqi Zhang, Xiaoyun He, and Shengmei Zhao, "Quantum classification algorithm with multi-class parallel training", Quantum Information Processing 21 10, 358 (2022).

[94] Masahito Hayashi and Yuxiang Yang, "Efficient algorithms for quantum information bottleneck", arXiv:2208.10342, (2022).

[95] Jonas Landman, Natansh Mathur, Yun Yvonna Li, Martin Strahm, Skander Kazdaghli, Anupam Prakash, and Iordanis Kerenidis, "Quantum Methods for Neural Networks and Application to Medical Image Classification", Quantum 6, 881 (2022).

[96] Bryan Liu, Toshiaki Koike-Akino, Ye Wang, and Kieran Parsons, "Variational Quantum Compressed Sensing for Joint User and Channel State Acquisition in Grant-Free Device Access Systems", arXiv:2205.08603, (2022).

[97] Sofiene Jerbi, Lukas J. Fiderer, Hendrik Poulsen Nautrup, Jonas M. Kübler, Hans J. Briegel, and Vedran Dunjko, "Quantum machine learning beyond kernel methods", Nature Communications 14, 517 (2023).

[98] Yong Siah Teo, Seongwook Shin, Hyukgun Kwon, Seok-Hyung Lee, and Hyunseok Jeong, "Virtual distillation with noise dilution", Physical Review A 107 2, 022608 (2023).

[99] Yong-Mei Li, Hai-Ling Liu, Shi-Jie Pan, Su-Juan Qin, Fei Gao, Dong-Xu Sun, and Qiao-Yan Wen, "Quantum k -medoids algorithm using parallel amplitude estimation", Physical Review A 107 2, 022421 (2023).

[100] Alexey Melnikov, Mohammad Kordzanganeh, Alexander Alodjants, and Ray-Kuang Lee, "Quantum machine learning: from physics to software engineering", Advances in Physics X 8 1, 2165452 (2023).

[101] Lucas Friedrich and Jonas Maziero, "Evolution strategies: application in hybrid quantum-classical neural networks", Quantum Information Processing 22 3, 132 (2023).

[102] Masahito Hayashi and Yuxiang Yang, "Efficient algorithms for quantum information bottleneck", Quantum 7, 936 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2022-09-13 01:15:33) and SAO/NASA ADS (last updated successfully 2023-03-20 10:30:28). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2023-03-20 10:30:23: Encountered the unhandled forward link type postedcontent_cite while looking for citations to DOI 10.22331/q-2020-02-06-226.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.