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.

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

[2] Lucas Friedrich and Jonas Maziero, "Avoiding barren plateaus with classical deep neural networks", Physical Review A 106 4, 042433 (2022).

[3] Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim A. Nicoli, Paolo Stornati, Rouven Koch, Miriam Büttner, Robert Okuła, Gorka Muñoz-Gil, Rodrigo A. Vargas-Hernández, Alba Cervera-Lierta, Juan Carrasquilla, Vedran Dunjko, Marylou Gabrié, Patrick Huembeli, Evert van Nieuwenburg, Filippo Vicentini, Lei Wang, Sebastian J. Wetzel, Giuseppe Carleo, Eliška Greplová, Roman Krems, Florian Marquardt, Michał Tomza, Maciej Lewenstein, and Alexandre Dauphin, "Modern applications of machine learning in quantum sciences", arXiv:2204.04198, (2022).

[4] David Peral García, Juan Cruz-Benito, and Francisco José García-Peñalvo, "Systematic Literature Review: Quantum Machine Learning and its applications", arXiv:2201.04093, (2022).

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

[6] 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).

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

[8] 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, 4919 (2022).

[9] 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).

[10] Johannes Jakob Meyer, Marian Mularski, Elies Gil-Fuster, Antonio Anna Mele, Francesco Arzani, Alissa Wilms, and Jens Eisert, "Exploiting Symmetry in Variational Quantum Machine Learning", PRX Quantum 4 1, 010328 (2023).

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

[12] El Amine Cherrat, Snehal Raj, Iordanis Kerenidis, Abhishek Shekhar, Ben Wood, Jon Dee, Shouvanik Chakrabarti, Richard Chen, Dylan Herman, Shaohan Hu, Pierre Minssen, Ruslan Shaydulin, Yue Sun, Romina Yalovetzky, and Marco Pistoia, "Quantum Deep Hedging", Quantum 7, 1191 (2023).

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

[14] 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, 161 (2021).

[15] Louis Schatzki, Martín Larocca, Quynh T. Nguyen, Frédéric Sauvage, and M. Cerezo, "Theoretical guarantees for permutation-equivariant quantum neural networks", npj Quantum Information 10, 12 (2024).

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

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

[18] 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, 702 (2022).

[19] 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).

[20] 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).

[21] Nicolas Heurtel, Andreas Fyrillas, Grégoire de Gliniasty, Raphaël Le Bihan, Sébastien Malherbe, Marceau Pailhas, Eric Bertasi, Boris Bourdoncle, Pierre-Emmanuel Emeriau, Rawad Mezher, Luka Music, Nadia Belabas, Benoît Valiron, Pascale Senellart, Shane Mansfield, and Jean Senellart, "Perceval: A Software Platform for Discrete Variable Photonic Quantum Computing", Quantum 7, 931 (2023).

[22] 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).

[23] A. Rizzo, N. Parmiggiani, A. Bulgarelli, A. Macaluso, V. Fioretti, L. Castaldini, A. Di Piano, G. Panebianco, C. Pittori, M. Tavani, C. Sartori, C. Burigana, V. Cardone, F. Farsian, M. Meneghetti, G. Murante, R. Scaramella, F. Schillirò, V. Testa, and T. Trombetti, "Quantum Convolutional Neural Networks for the detection of Gamma-Ray Bursts in the AGILE space mission data", arXiv:2404.14133, (2024).

[24] 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).

[25] Elena Peña Tapia, Giannicola Scarpa, and Alejandro Pozas-Kerstjens, "A didactic approach to quantum machine learning with a single qubit", Physica Scripta 98 5, 054001 (2023).

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

[27] Youle Wang, Lei Zhang, Zhan Yu, and Xin Wang, "Quantum phase processing and its applications in estimating phase and entropies", Physical Review A 108 6, 062413 (2023).

[28] 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).

[29] Evan Peters and Maria Schuld, "Generalization despite overfitting in quantum machine learning models", Quantum 7, 1210 (2023).

[30] Aleksei Tolstobrov, Gleb Fedorov, Shtefan Sanduleanu, Shamil Kadyrmetov, Andrei Vasenin, Aleksey Bolgar, Daria Kalacheva, Viktor Lubsanov, Aleksandr Dorogov, Julia Zotova, Peter Shlykov, Aleksei Dmitriev, Konstantin Tikhonov, and Oleg V. Astafiev, "Hybrid quantum learning with data reuploading on a small-scale superconducting quantum simulator", Physical Review A 109 1, 012411 (2024).

[31] Vasilis Belis, Patrick Odagiu, and Thea Klæboe Årrestad, "Machine Learning for Anomaly Detection in Particle Physics", arXiv:2312.14190, (2023).

[32] Sebastián Roca-Jerat, Juan Román-Roche, and David Zueco, "Qudit machine learning", Machine Learning: Science and Technology 5 1, 015057 (2024).

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

[34] Yuxuan Du, Yibo Yang, Dacheng Tao, and Min-Hsiu Hsieh, "Problem-Dependent Power of Quantum Neural Networks on Multiclass Classification", Physical Review Letters 131 14, 140601 (2023).

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

[36] Franz J. Schreiber, Jens Eisert, and Johannes Jakob Meyer, "Classical Surrogates for Quantum Learning Models", Physical Review Letters 131 10, 100803 (2023).

[37] Elies Gil-Fuster, Jens Eisert, and Carlos Bravo-Prieto, "Understanding quantum machine learning also requires rethinking generalization", Nature Communications 15, 2277 (2024).

[38] Stavros Efthymiou, Alvaro Orgaz-Fuertes, Rodolfo Carobene, Juan Cereijo, Andrea Pasquale, Sergi Ramos-Calderer, Simone Bordoni, David Fuentes-Ruiz, Alessandro Candido, Edoardo Pedicillo, Matteo Robbiati, Yuanzheng Paul Tan, Jadwiga Wilkens, Ingo Roth, José Ignacio Latorre, and Stefano Carrazza, "Qibolab: an open-source hybrid quantum operating system", Quantum 8, 1247 (2024).

[39] 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).

[40] Dar Gilboa and Jarrod R. McClean, "Exponential Quantum Communication Advantage in Distributed Learning", arXiv:2310.07136, (2023).

[41] Jingwei Wen, Zhiguo Huang, Dunbo Cai, and Ling Qian, "Enhancing the expressivity of quantum neural networks with residual connections", arXiv:2401.15871, (2024).

[42] Annie E. Paine, Vincent E. Elfving, and Oleksandr Kyriienko, "Quantum kernel methods for solving regression problems and differential equations", Physical Review A 107 3, 032428 (2023).

[43] 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).

[44] 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).

[45] Mo Kordzanganeh, Pavel Sekatski, Leonid Fedichkin, and Alexey Melnikov, "An exponentially-growing family of universal quantum circuits", Machine Learning: Science and Technology 4 3, 035036 (2023).

[46] 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).

[47] Berta Casas and Alba Cervera-Lierta, "Multidimensional Fourier series with quantum circuits", Physical Review A 107 6, 062612 (2023).

[48] Seongwook Shin, Yong Siah Teo, and Hyunseok Jeong, "Dequantizing quantum machine learning models using tensor networks", arXiv:2307.06937, (2023).

[49] Smit Chaudhary, Patrick Huembeli, Ian MacCormack, Taylor L. Patti, Jean Kossaifi, and Alexey Galda, "Towards a scalable discrete quantum generative adversarial neural network", Quantum Science and Technology 8 3, 035002 (2023).

[50] Zhongtian Dong, Marçal Comajoan Cara, Gopal Ramesh Dahale, Roy T. Forestano, Sergei Gleyzer, Daniel Justice, Kyoungchul Kong, Tom Magorsch, Konstantin T. Matchev, Katia Matcheva, and Eyup B. Unlu, "$\mathbb{Z}_2\times \mathbb{Z}_2$ Equivariant Quantum Neural Networks: Benchmarking against Classical Neural Networks", arXiv:2311.18744, (2023).

[51] Elies Gil-Fuster, Jens Eisert, and Vedran Dunjko, "On the expressivity of embedding quantum kernels", Machine Learning: Science and Technology 5 2, 025003 (2024).

[52] 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).

[53] Ben Jaderberg, Antonio A. Gentile, Youssef Achari Berrada, Elvira Shishenina, and Vincent E. Elfving, "Let quantum neural networks choose their own frequencies", Physical Review A 109 4, 042421 (2024).

[54] 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).

[55] Bojia Duan, Xin Sun, and Chang-Yu Hsieh, "Parallelized variational quantum classifier with shallow QRAM circuit", Quantum Information Processing 23 3, 92 (2024).

[56] Chih-Hung Wu and Ching-Che Yen, "The expressivity of classical and quantum neural networks on entanglement entropy", European Physical Journal C 84 2, 192 (2024).

[57] 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).

[58] Hela Mhiri, Leo Monbroussou, Mario Herrero-Gonzalez, Slimane Thabet, Elham Kashefi, and Jonas Landman, "Constrained and Vanishing Expressivity of Quantum Fourier Models", arXiv:2403.09417, (2024).

[59] Yingli Yang, Zongkang Zhang, Anbang Wang, Xiaosi Xu, Xiaoting Wang, and Ying Li, "Maximizing quantum-computing expressive power through randomized circuits", Physical Review Research 6 2, 023098 (2024).

[60] Pablo Bermejo and Román Orús, "Variational quantum and quantum-inspired clustering", Scientific Reports 13, 13284 (2023).

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

[62] 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).

[63] Rodrigo Martínez-Peña and Juan-Pablo Ortega, "Quantum reservoir computing in finite dimensions", Physical Review E 107 3, 035306 (2023).

[64] Y. S. Teo, "Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms", Physical Review A 109 1, 012620 (2024).

[65] 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).

[66] Xiaokai Hou, Guanyu Zhou, Qingyu Li, Shan Jin, and Xiaoting Wang, "A duplication-free quantum neural network for universal approximation", Science China Physics, Mechanics, and Astronomy 66 7, 270362 (2023).

[67] Shaozhi Li, M Sabbir Salek, Yao Wang, and Mashrur Chowdhury, "Quantum-inspired activation functions in the convolutional neural network", arXiv:2404.05901, (2024).

[68] Anqi Zhang and Shengmei Zhao, "Evolutionary-based searching method for quantum circuit architecture", Quantum Information Processing 22 7, 283 (2023).

[69] Chuan-Dong Song, Jian Li, Yan-Yan Hou, Qing-Hui Liu, and Zhuo Wang, "Quantum canonical correlation analysis algorithm", Laser Physics Letters 20 10, 105203 (2023).

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

[71] 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).

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

[73] Erfan Abedi, Salman Beigi, and Leila Taghavi, "Quantum Lazy Training", Quantum 7, 989 (2023).

[74] 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, 11927 (2022).

[75] Marco Ballarin, Stefano Mangini, Simone Montangero, Chiara Macchiavello, and Riccardo Mengoni, "Entanglement entropy production in Quantum Neural Networks", Quantum 7, 1023 (2023).

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

[77] 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).

[78] Mahsa Karimi, Ali Javadi-Abhari, Christoph Simon, and Roohollah Ghobadi, "The power of one clean qubit in supervised machine learning", Scientific Reports 13, 19975 (2023).

[79] Javier Mancilla and Christophe Pere, "A Preprocessing Perspective for Quantum Machine Learning Classification Advantage in Finance Using NISQ Algorithms", Entropy 24 11, 1656 (2022).

[80] 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).

[81] Chih-Chieh Chen, Masaru Sogabe, Kodai Shiba, Katsuyoshi Sakamoto, and Tomah Sogabe, "General Vapnik-Chervonenkis dimension bounds for quantum circuit learning", Journal of Physics: Complexity 3 4, 045007 (2022).

[82] Yudai Suzuki, Rei Sakuma, and Hideaki Kawaguchi, "Light-cone feature selection for quantum machine learning", arXiv:2403.18733, (2024).

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

[84] Yue Ban, E. Torrontegui, and J. Casanova, "Quantum neural networks with multi-qubit potentials", Scientific Reports 13, 9096 (2023).

[85] Soronzonbold Otgonbaatar, Gottfried Schwarz, Mihai Datcu, and Dieter Kranzlmüller, "Quantum Transfer Learning for Real-World, Small, and High-Dimensional Remotely Sensed Datasets", IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing 16, 9223 (2023).

[86] Lucas Friedrich and Jonas Maziero, "Restricting to the chip architecture maintains the quantum neural network accuracy", Quantum Information Processing 23 4, 131 (2024).

[87] 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).

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

[89] 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).

[90] Li Ding and Lee Spector, "Multi-Objective Evolutionary Architecture Search for Parameterized Quantum Circuits", Entropy 25 1, 93 (2023).

[91] Yudai Suzuki and Muyuan Li, "Effect of alternating layered ansatzes on trainability of projected quantum kernel", arXiv:2310.00361, (2023).

[92] Pablo Bermejo and Román Orús, "Variational quantum non-orthogonal optimization", Scientific Reports 13, 9840 (2023).

[93] Vasilis Belis, Patrick Odagiu, Michele Grossi, Florentin Reiter, Günther Dissertori, and Sofia Vallecorsa, "Guided Quantum Compression for Higgs Identification", arXiv:2402.09524, (2024).

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

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

[96] Eva Andrés, M. P. Cuéllar, and G. Navarro, "Efficient Dimensionality Reduction Strategies for Quantum Reinforcement Learning", IEEE Access 11, 104534 (2023).

[97] Nico Meyer, Daniel D. Scherer, Axel Plinge, Christopher Mutschler, and Michael J. Hartmann, "Quantum Policy Gradient Algorithm with Optimized Action Decoding", arXiv:2212.06663, (2022).

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

[99] Y. S. Teo, "Optimized numerical gradient and Hessian estimation for variational quantum algorithms", Physical Review A 107 4, 042421 (2023).

[100] Axel Pérez-Obiol, Adrián Pérez-Salinas, Sergio Sánchez-Ramírez, Bruna G. M. Araújo, and Artur Garcia-Saez, "Adiabatic quantum algorithm for artificial graphene", Physical Review A 106 5, 052408 (2022).

[101] Hanif Heidari, Gerhard Hellstern, and Murugappan Murugappan, "Heart Disease Detection using Quantum Computing and Partitioned Random Forest Methods", arXiv:2208.08882, (2022).

[102] Takafumi Ono, Wojciech Roga, Kentaro Wakui, Mikio Fujiwara, Shigehito Miki, Hirotaka Terai, and Masahiro Takeoka, "Demonstration of a Bosonic Quantum Classifier with Data Reuploading", Physical Review Letters 131 1, 013601 (2023).

[103] Iván Panadero, Yue Ban, Hilario Espinós, Ricardo Puebla, Jorge Casanova, and Erik Torrontegui, "Regressions on quantum neural networks at maximal expressivity", arXiv:2311.06090, (2023).

[104] 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).

[105] Yong-Mei Li, Hai-Ling Liu, Shi-Jie Pan, Su-Juan Qin, Fei Gao, and Qiao-Yan Wen, "Quantum discriminative canonical correlation analysis", Quantum Information Processing 22 4, 163 (2023).

[106] A. Mandilara, B. Dellen, U. Jaekel, T. Valtinos, and D. Syvridis, "Classification of data with a qudit, a geometric approach", arXiv:2307.14060, (2023).

[107] Ruhan Wang, Philip Richerme, and Fan Chen, "A hybrid quantum-classical neural network for learning transferable visual representation", Quantum Science and Technology 8 4, 045021 (2023).

[108] Tailong Xiao, Jingzheng Huang, Hongjing Li, Jianping Fan, and Guihua Zeng, "Quantum generative adversarial imitation learning", New Journal of Physics 25 3, 033034 (2023).

[109] Elijah Pelofske, "Single Qubit Multi-Party Transmission Using Universal Symmetric Quantum Cloning", arXiv:2310.04920, (2023).

[110] Nikolaos Schetakis, Davit Aghamalyan, Michael Boguslavsky, Agnieszka Rees, Marc Raktomalala, and Paul Griffin, "Quantum Machine Learning for Credit Scoring", arXiv:2308.03575, (2023).

[111] Stefano Markidis, "Programming Quantum Neural Networks on NISQ Systems: An Overview of Technologies and Methodologies", Entropy 25 4, 694 (2023).

[112] Paul San Sebastian, Mikel Cañizo, and Román Orús, "Image Classification with Rotation-Invariant Variational Quantum Circuits", arXiv:2403.15031, (2024).

[113] Michael Kölle, Alessandro Giovagnoli, Jonas Stein, Maximilian Balthasar Mansky, Julian Hager, Tobias Rohe, Robert Müller, and Claudia Linnhoff-Popien, "Weight Re-Mapping for Variational Quantum Algorithms", arXiv:2306.05776, (2023).

[114] 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).

[115] Nannan Ma, Wenhao Chu, P. Z. Zhao, and Jiangbin Gong, "Adiabatic quantum learning", Physical Review A 108 4, 042420 (2023).

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

[117] Maureen Monnet, Nermine Chaabani, Theodora-Augustina Dragan, Balthasar Schachtner, and Jeanette Miriam Lorenz, "Understanding the effects of data encoding on quantum-classical convolutional neural networks", arXiv:2405.03027, (2024).

[118] 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).

[119] Mushrafi Munim Sushmit and Islam Mohammed Mahbubul, "Forecasting solar irradiance with hybrid classical-quantum models: A comprehensive evaluation of deep learning and quantum-enhanced techniques", Energy Conversion and Management 294, 117555 (2023).

[120] Stian Bilek, "Quantum Machine Learning With a Limited Number Of Qubits", arXiv:2403.14406, (2024).

[121] Yuan Li and Jin-Yang Li, "Quantum Coding via Quasi-Cyclic Block Matrix", Entropy 25 3, 537 (2023).

[122] Maida Shahid and Muhammad Awais Hassan, "Introducing Quantum Variational Circuit for Efficient Management of Common Pool Resources", IEEE Access 11, 110862 (2023).

[123] Alexander Benítez-Buenache and Queralt Portell-Montserrat, "Bayesian Parameterized Quantum Circuit Optimization (BPQCO): A task and hardware-dependent approach", arXiv:2404.11253, (2024).

[124] Bowen Li, Ting Li, and Fei Li, "A design method for efficient variational quantum models based on specific Pauli axis", Quantum Information Processing 22 10, 387 (2023).

[125] E. F. Combarro, I. F. Rúa, F. Orts, G. Ortega, A. M. Puertas, and E. M. Garzón, "Quantum algorithms to compute the neighbour list of N-body simulations", Quantum Information Processing 23 2, 61 (2024).

[126] Lian-Hui Yu, Xiao-Yu Li, Geng Chen, Qin-Sheng Zhu, and Guo-Wu Yang, "QUSL: Quantum Unsupervised Image Similarity Learning with Enhanced Performance", arXiv:2404.02028, (2024).

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