Skip to main content

Enabling Non-linear Quantum Operations Through Variational Quantum Splines

  • Conference paper
  • First Online:
Computational Science – ICCS 2023 (ICCS 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14077))

Included in the following conference series:

Abstract

One of the major issues for building a complete quantum neural network is the implementation of non-linear activation functions in a quantum computer. In fact, the postulates of quantum mechanics impose only unitary transformations on quantum states, which is a severe limitation for quantum machine learning algorithms. Recently, the idea of QSplines has been proposed to approximate non-linear quantum activation functions by means of the HHL. However, QSplines rely on a problem formulation to be represented as a block diagonal matrix and need a fault-tolerant quantum computer to be correctly implemented.

This work proposes two novel methods for approximating non-linear quantum activation functions using variational quantum algorithms. Firstly, we develop the variational QSplines (VQSplines) that allow overcoming the highly demanding requirements of the original QSplines and approximating non-linear functions using near-term quantum computers. Secondly, we propose a novel formulation for QSplines, the Generalized QSplines (GQSplines), which provide a more flexible representation of the problem and are suitable to be embedded in existing quantum neural network architectures. As a third meaningful contribution, we implement VQSplines and GQSplines using Pennylane to show the effectiveness of the proposed approaches in approximating typical non-linear activation functions in a quantum computer.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It is also possible to define the Hermitian matrix H from S as \(H=\begin{pmatrix} 0 &{} S \\ S^\dag &{} 0 \end{pmatrix}\).

  2. 2.

    Though the primary objective of QSplines is to embed non-linearity into quantum neural networks, they can serve to approximate other types of non-linearity.

  3. 3.

    https://pennylane.ai/.

References

  1. Yuan, X.: A quantum-computing advantage for chemistry. Science 369(6507), 1054–1055 (2020)

    Article  Google Scholar 

  2. Venkatesh, S.M., Macaluso, A., Klusch, M.: BILP-Q: quantum coalition structure generation. In: Proceedings of the 19th ACM International Conference on Computing Frontiers, pp. 189–192 (2022)

    Google Scholar 

  3. Venkatesh, S.M., Macaluso, A., Klusch, M.: GCS-Q: quantum graph coalition structure generation. arXiv preprint arXiv:2212.11372 (2022)

  4. Macaluso, A., Clissa, L., Lodi, S., Sartori, C.: Quantum splines for non-linear approximations. In: Proceedings of the 17th ACM International Conference on Computing Frontiers, CF 2020, pp. 249–252, New York, USA, Association for Computing Machinery (2020)

    Google Scholar 

  5. Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 10 (2009)

    Article  MathSciNet  Google Scholar 

  6. Cao, Y., Guerreschi, G.G., Aspuru-Guzik, A.: Quantum neuron: an elementary building block for machine learning on quantum computers

    Google Scholar 

  7. Maronese, M., Destri, C., Prati, E.: Quantum activation functions for quantum neural networks (2022)

    Google Scholar 

  8. Lubasch, M., Joo, J., Moinier, P., Kiffner, M., Jaksch, D.: Variational quantum algorithms for nonlinear problems. Phys. Rev. A 101, 010301 (2020)

    Article  Google Scholar 

  9. Bravo-Prieto, C., LaRose, R., Cincio, M.L., Coles, P.J.: Variational quantum linear solver, Cerezo, Yigit Subasi (2019)

    Google Scholar 

  10. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. SSS, Springer, New York (2009). https://doi.org/10.1007/978-0-387-84858-7

    Book  Google Scholar 

  11. de Boor, C.: A Practical Guide to Splines. Springer Verlag, New York (1978)

    Google Scholar 

  12. Buhrman, H., Cleve, R., Watrous, J., de Wolf, R.: Quantum fingerprinting. Phys. Rev. Lett. 87, 167902 (2001)

    Article  Google Scholar 

  13. Markov, V., Stefanski, C., Rao, A., Gonciulea, C.: A generalized quantum inner product and applications to financial engineering. arXiv preprint arXiv:2201.09845 (2022)

  14. Mottonen, M., Vartiainen, J.J.: Decompositions of general quantum gates. Ch. 7 in Trends in Quantum Computing Research, NOVA Publishers, New York, 2006 (2005)

    Google Scholar 

  15. Cleve, R., Ekert, A., Macchiavello, C., Mosca, M.: Quantum algorithms revisited. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 454(1969), 339–354 (1998)

    Google Scholar 

  16. Rice, J.R.: A theory of condition. SIAM J. Numer. Anal. 3(2), 287–310 (1966)

    Article  MathSciNet  Google Scholar 

  17. Brassard, G., Hoyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. Quantum Comput. Quantum Inf. 305, 53–74 (2000)

    Google Scholar 

  18. Macaluso, A., Clissa, L., Lodi, S., Sartori, C.: A variational algorithm for quantum neural networks. In: Krzhizhanovskaya, V.V. (ed.) ICCS 2020. LNCS, vol. 12142, pp. 591–604. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-50433-5_45

    Chapter  Google Scholar 

  19. Macaluso, A., Orazi, F., Klusch, M., Lodi, S., Sartori, C.: A variational algorithm for quantum single layer perceptron. In: , et al. Machine Learning, Optimization, and Data Science. LOD 2022. Lecture Notes in Computer Science. vol. 13811. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-25891-6_26

  20. Macaluso, A., Klusch, M., Lodi, S., et al.: MAQA: a quantum framework for supervised learning. Quantum Inf. Process. 22, 159 (2023). https://doi.org/10.1007/s11128-023-03901-w

  21. Barenco, A., Berthiaume, A., Deutsch, D., Ekert, A., Jozsa, R., Macchiavello, C.: Stabilization of quantum computations by symmetrization. SIAM J. Comput. 26(5), 1541–1557 (1997)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has been partially funded by the German Ministry for Education and Research (BMB+F) in the project QAI2-QAICO under grant 13N15586.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Filippo Orazi .

Editor information

Editors and Affiliations

Ethics declarations

Code Availability

All code to generate the data, figures, analyses, and additional details on the experiments are available at https://github.com/inajetovic/Variational-Quantum-Splines.

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Inajetovic, M.A., Orazi, F., Macaluso, A., Lodi, S., Sartori, C. (2023). Enabling Non-linear Quantum Operations Through Variational Quantum Splines. In: Mikyška, J., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M. (eds) Computational Science – ICCS 2023. ICCS 2023. Lecture Notes in Computer Science, vol 14077. Springer, Cham. https://doi.org/10.1007/978-3-031-36030-5_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-36030-5_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-36029-9

  • Online ISBN: 978-3-031-36030-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics