Data re-uploading for a universal quantum classifier

Adrián Pérez-Salinas1,2, Alba Cervera-Lierta1,2, Elies Gil-Fuster3, and José I. Latorre1,2,4,5

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.

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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


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.

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.

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► References

[1] M. Schuld, I. Sinayskiy, and F. Petruccione, Quantum Information Processing 13, 2567 (2014).

[2] K. H. Wan, O. Dahlsten, H. Kristjánsson, R. Gardner, and M. S. Kim, npj Quantum Information 3, 36 (2017).

[3] E. Torrontegui and J. J. García-Ripoll, EPL (Europhysics Letters) 125, 30004 (2019).

[4] N. Wiebe, D. Braun, and S. Lloyd, Physics Review Letters 109, 050505 (2012).

[5] P. Rebentrost, M. Mohseni, and S. Lloyd, Physics Review Letters 113, 130503 (2014).

[6] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature 549, 195 (2017).

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

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

[9] V. Havlíček, A. D. Córcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, Nature 567, 209 (2019).

[10] M. Schuld and N. Killoran, Physics Review Letters 122, 040504 (2019).

[11] V. Giovannetti, S. Lloyd, and L. Maccone, Physics Review Letters 100, 160501 (2008).

[12] K. Hornik, Neural Networks 4, 251 (1991).

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

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

[15] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Physics Review A 98, 032309 (2018).

[16] C. W. Helstrom, Quantum detection and estimation theory /​ Carl W. Helstrom (Academic Press New York, 1976) pp. ix, p. : 309.

[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).

[18] G. Cybenko, Mathematics of Control, Signals, and Systems 2, 303 (1989).

[19] B. C. Hall, Lie Groups, Lie Algebras, and Representations An Elementary Introduction (Graduate Texts in Mathematics, 222 (2nd ed.), Springer, 2015).

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

[21] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, SIAM Journal on Scientific Computing 16, 1190 (1995).

[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).

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

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

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

[26] J. Romero, R. Babbush, J. R. McClean, C. Hempel, P. J. Love, and A. Aspuru-Guzik, Quantum Science and Technology 4, 014008 (2018).

Cited by

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

[2] 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", arXiv:2303.16585, (2023).

[3] 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).

[4] 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).

[5] 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).

[6] 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).

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

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

[9] 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).

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

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

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

[13] 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).

[14] 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).

[15] 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).

[16] 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).

[17] 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).

[18] 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).

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

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

[21] 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).

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

[23] 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).

[24] 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).

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

[26] 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).

[27] 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).

[28] 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).

[29] 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).

[30] 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).

[31] 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).

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

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

[34] 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).

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

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

[37] 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).

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

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

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

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

[42] 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).

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

[44] 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).

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

[46] 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).

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

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

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

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

[51] 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).

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

[53] Hanif Heidari and Gerhard Hellstern, "Early heart disease prediction using hybrid quantum classification", arXiv:2208.08882, (2022).

[54] 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).

[55] 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).

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

[57] 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).

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

[59] 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).

[60] 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).

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

[62] 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).

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

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

[65] 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).

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

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

[68] Soronzonbold Otgonbaatar, Gottfried Schwarz, Mihai Datcu, and Dieter Kranzlmüller, "Quantum Transfer Learning for Real-World, Small, and High-Dimensional Datasets", arXiv:2209.07799, (2022).

[69] 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).

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

[71] 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).

[72] 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).

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

[74] 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).

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