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Universal approximation of continuous functions with minimal quantum circuits

Adrián Pérez-Salinas1,2,*, Mahtab Yaghubi Rad1,3, Alice Barthe1,4, and Vedran Dunjko1,3

  • *Contact author: adperez@phys.ethz.ch
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Phys. Rev. Research 7, 043282 – Published 10 December, 2025

DOI: https://doi.org/10.1103/t49h-jmty

Abstract

The conventional paradigm of quantum computing is discrete: It utilizes discrete sets of gates to realize bitstring-to-bitstring mappings, some of them arguably intractable for classical computers. In parametrized quantum approaches, the input becomes continuous and the output represents real-valued functions. While the universality of discrete quantum computers is well understood, basic questions remained open in the continuous case. We focus on universality of multivariate functions. Current approaches require either a number of qubits scaling linearly with the dimension of the input for fixed encodings or a tunable encoding procedure in single-qubit circuits. The question of whether universality can be reached with a fixed encoding and sublinearly many qubits remained open for the last five years. In this paper, we answer this question in the affirmative for arbitrary multivariate functions. We provide two methods: (1) a single-qubit circuit where each coordinate of the arguments to the function to represent is input independently and (2) a multiqubit approach where all coordinates are input in one step, with number of qubits scaling logarithmically with the dimension of the argument of the function of interest. We view the first result of inherent and fundamental interest, whereas the second result opens the path toward representing functions whose arguments are densely encoded in a unitary operation, possibly encoding, for instance, quantum processes.

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References (28)

  1. P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM J. Comput. 26, 1484 (1997).
  2. A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, A variational eigenvalue solver on a photonic quantum processor, Nat. Commun. 5, 4213 (2014).
  3. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  4. A. Cervera-Lierta, J. S. Kottmann, and A. Aspuru-Guzik, The meta-variational quantum eigensolver (meta-VQE): Learning energy profiles of parameterized Hamiltonians for quantum simulation, PRX Quantum 2, 020329 (2021).
  5. V. Havlíček, A. D. Córcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, Supervised learning with quantum-enhanced feature spaces, Nature (London) 567, 209 (2019).
  6. M. Schuld and N. Killoran, Quantum machine learning in feature Hilbert spaces, Phys. Rev. Lett. 122, 040504 (2019).
  7. K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Quantum circuit learning, Phys. Rev. A 98, 032309 (2018).
  8. 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).
  9. In the quantum case, we also have the universality of (approximately) representing arbitrary unitary transformations, which is also guaranteed for any so-called universal gate set, but this is tangential to our discussion.
  10. H. M. Sheffer, A set of five independent postulates for Boolean algebras, with application to logical constants, Trans. Am. Math. Soc. 14, 481 (1913).
  11. D. A. Barrington, Bounded-width polynomial-size branching programs recognize exactly those languages in NC1, J. Comput. Syst. Sci. 38, 150 (1989).
  12. A. Cosentino, R. Kothari, and A. Paetznick, Dequantizing read-once quantum formulas, in 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013) (Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2013), pp. 80–92.
  13. G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2, 303 (1989).
  14. K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Netw. 4, 251 (1991).
  15. V. I. Arnold, On the representation of functions of several variables as a superposition of functions of a smaller number of variables, in Collected Works: Representations of Functions, Celestial Mechanics and KAM Theory, 1957–1965, edited by V. I. Arnold, A. B. Givental, B. A. Khesin, J. E. Marsden, A. N. Varchenko, V. A. Vassiliev, O. Y. Viro, and V. M. Zakalyukin (Springer, Berlin, 2009), pp. 25–46.
  16. Z. Yu, Q. Chen, Y. Jiao, Y. Li, X. Lu, X. Wang, and J. Z. Yang, Non-asymptotic approximation error bounds of parameterized quantum circuits, Adv. Neural Inf. Process. Syst. 37, 99089 (2024).
  17. M. Schuld, R. Sweke, and J. J. Meyer, The effect of data encoding on the expressive power of variational quantum machine learning models, Phys. Rev. A 103, 032430 (2021).
  18. D. Motlagh and N. Wiebe, Generalized quantum signal processing, PRX Quantum 5, 020368 (2024).
  19. T. d. L. Silva, L. Borges, and L. Aolita, Fourier-based quantum signal processing, arXiv:2206.02826.
  20. In this work, we will use the words parameters and arguments for making the difference between θ and x explicit, respectively.
  21. B. Casas and A. Cervera-Lierta, Multidimensional Fourier series with quantum circuits, Phys. Rev. A 107, 062612 (2023).
  22. 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).
  23. P. G. L. Dirichlet, Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données, arXiv:0806.1294.
  24. P. Turán, Sur Les Fonctions Bornées et Intégrables, in Leopold Fejér Gesammelte Arbeiten I, edited by P. Turán (Birkhäuser, Basel, 1970), pp. 37–43.
  25. M. Arató and A. Rényi, Probabilistic proof of a theorem on the approximation of continuous functions by means of generalized Bernstein polynomials, Acta Math. Acad. Sci. Hung. 8, 91 (1957).
  26. A. Manzano, D. Dechant, J. Tura, and V. Dunjko, Parametrized quantum circuits and their approximation capacities in the context of quantum machine learning, Quantum 9, 1658 (2023).
  27. G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).
  28. T. Tilma and E. C. G. Sudarshan, Generalized Euler angle paramterization for SU(N), J. Phys. A: Math. Gen. 35, 10467 (2002).

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Phys. Rev. Research 7, 043282– Published 10 December, 2025
  • Received 10 May 2025
  • Accepted 27 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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