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.

### 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.

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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### Cited by

[1] 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).

[2] Patrick Huembeli and Alexandre Dauphin, "Characterizing the loss landscape of variational quantum circuits", arXiv:2008.02785, Quantum Science and Technology 6 2, 025011 (2021).

[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] 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.

[5] 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).

[6] 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).

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

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

[9] Teppei Suzuki and Michio Katouda, "Predicting toxicity by quantum machine learning", Journal of Physics Communications 4 12, 125012 (2021).

[10] 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).

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

[12] William Cappelletti, Rebecca Erbanni, and Joaquín Keller, "Polyadic Quantum Classifier", arXiv:2007.14044.

[13] 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.

The above citations are from Crossref's cited-by service (last updated successfully 2021-03-06 08:18:03) and SAO/NASA ADS (last updated successfully 2021-03-06 08:18:04). The list may be incomplete as not all publishers provide suitable and complete citation data.