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Multidimensional Fourier series with quantum circuits
Phys. Rev. A 107, 062612 – Published 29 June, 2023
DOI: https://doi.org/10.1103/PhysRevA.107.062612
Abstract
Quantum machine learning is the field that aims to integrate machine learning with quantum computation. In recent years, the field has emerged as an active research area with the potential to bring new insights to classical machine learning problems. One of the challenges in the field is to explore the expressibility of parametrized quantum circuits and their ability to be universal function approximators, as classical neural networks are. Recent works have shown that, with a quantum supervised learning model, we can fit any one-dimensional Fourier series, proving their universality. However, models for multidimensional functions have not been explored in the same level of detail. In this work, we study the expressibility of various types of circuit Ansätze that generate multidimensional Fourier series. We found that, for some Ansätze, the degrees of freedom required for fitting such functions grow faster than the available degrees in the Hilbert space generated by the circuits. For example, single-qudit models have limited power to represent arbitrary multidimensional Fourier series. Despite this, we show that we can enlarge the Hilbert space of the circuit by using more qudits or higher local dimensions to meet the degrees of freedom requirements, thus ensuring the universality of the models.
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References (30)
- J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature (London) 549, 195 (2017).
- A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Data re-uploading for a universal quantum classifier, Quantum 4, 226 (2020).
- S. Lloyd, M. Schuld, A. Ijaz, J. Izaac, and N. Killoran, Quantum embeddings for machine learning, arXiv:2001.03622.
- T. Goto, Q. H. Tran, and K. Nakajima, Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces, Phys. Rev. Lett. 127, 090506 (2021).
- F. J. G. Vidal and D. O. Theis, Input redundancy for parameterized quantum circuits, Front. Phys. 8, 297 (2020).
- M. Schuld, R. Sweke, and J. J. Meyer, Effect of data encoding on the expressive power of variational quantum-machine-learning models, Phys. Rev. A 103, 032430 (2021).
- D. Heimann, G. Schönhoff, and F. Kirchner, Learning capability of parametrized quantum circuits, arXiv:2209.10345.
- S. Shin, Y. S. Teo, and H. Jeong, Exponential data encoding for quantum supervised learning, Phys. Rev. A 107, 012422 (2023).
- A. Daskin, A walk through of time series analysis on quantum computers, arXiv:2205.00986.
- Z. Yu, H. Yao, M. Li, and X. Wang, Power and limitations of single-qubit native quantum neural networks, arXiv:2205.07848.
- B. Y. Gan, D. Leykam, and D. G. Angelakis, Fock state-enhanced expressivity of quantum machine learning models, EPJ Quantum Technol. 9, 16 (2022).
- K. A. Woźniak, V. Belis, E. Puljak, P. Barkoutsos, G. Dissertori et al., Quantum anomaly detection in the latent space of proton collision events at the lhc, arXiv:2301.10780.
- J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
- K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W. K. Mok, S. Sim, L. C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys. 94, 015004 (2022).
- M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo et al., Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021).
- M. S. Blok, V. V. Ramasesh, T. Schuster, K. O'Brien, J. M. Kreikebaum, D. Dahlen, A. Morvan, B. Yoshida, N. Y. Yao, and I. Siddiqi, Quantum Information Scrambling on a Superconducting Qutrit Processor, Phys. Rev. X 11, 021010 (2021).
- A. Cervera-Lierta, M. Krenn, A. Aspuru-Guzik, and A. Galda, Experimental High-Dimensional Greenberger-Horne-Zeilinger Entanglement with Superconducting Transmon Qutrits, Phys. Rev. Appl. 17, 024062 (2022).
- M. Erhard, M. Krenn, and A. Zeilinger, Advances in high-dimensional quantum entanglement, Nat. Rev. Phys. 2, 365 (2020).
- A. Castro, A. Garcia Carrizo, S. Roca, D. Zueco, and F. Luis, Optimal Control of Molecular Spin Qudits, Phys. Rev. Appl. 17, 064028 (2022).
- M. D. Jenkins, Y. Duan, B. Diosdado, J. J. Garcia-Ripoll, A. Gaita-Arino, C. Gimenez-Saiz, P. J. Alonso, E. Coronado, and F. Luis, Coherent manipulation of three-qubit states in a molecular single-ion magnet, Phys. Rev. B 95, 064423 (2017).
- C. M. Caves and G. J. Milburn, Qutrit entanglement, Opt. Commun. 179, 439 (2000).
- A. Pérez-Salinas, D. López-Núñez, A. García-Sáez, P. Forn-Díaz, and J. I. Latorre, One qubit as a universal approximant, Phys. Rev. A 104, 012405 (2021).
- V. Vapnik, E. Levin, and Y. L. Cun, Measuring the vc-dimension of a learning machine, Neural Comput. 6, 851 (1994).
- A. Canatar, E. Peters, C. Pehlevan, S. M. Wild, and R. Shaydulin, Bandwidth enables generalization in quantum kernel models, arXiv:2206.06686.
- A. Gratsea and P. Huembeli, The effect of the processing and measurement operators on the expressive power of quantum models, arXiv:2211.03101.
- J. Kübler, S. Buchholz, and B. Schölkopf, The inductive bias of quantum kernels, Adv. Neural Inf. Process. Syst. 34, 12661 (2021).
- E. Peters and M. Schuld, Generalization despite overfitting in quantum machine learning models, arXiv:2209.05523.
- F. J. Schreiber, J. Eisert, and J. J. Meyer, Classical surrogates for quantum learning models, arXiv:2206.11740.
- J. Landman, S. Thabet, C. Dalyac, H. Mhiri, and E. Kashefi, Classically approximating variational quantum machine learning with random Fourier features, arXiv:2210.13200.
- https://github.com/bsc-quantic/fourier.
