- Physical Review Journals
All JournalsPhysics Magazine

Physical Review A

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Editors' Suggestion
  • Letter
  • Open Access

Universal variational quantum computation

Jacob Biamonte*

  • Skolkovo Institute of Science and Technology, 3 Nobel Street, Moscow 121205, Russia

PDF

Phys. Rev. A 103, L030401 – Published 10 March, 2021

DOI: https://doi.org/10.1103/PhysRevA.103.L030401

Abstract

Variational quantum algorithms dominate contemporary gate-based quantum enhanced optimization, eigenvalue estimation, and machine learning. Here we establish the quantum computational universality of variational quantum computation by developing two objective functions which minimize to prepare outputs of arbitrary quantum circuits. The fleeting resource of variational quantum computation is the number of expected values which must be iteratively minimized using classical-to-quantum outer loop optimization. An efficient solution to this optimization problem is given by the quantum circuit being simulated itself. The first construction is efficient in the number of expected values for n-qubit circuits containing O(polylnn) non-Clifford gates—the number of expected values has no dependence on Clifford gates appearing in the simulated circuit. The second approach yields O(L2) expected values whereas introducing not more than O(lnL) slack qubits for a quantum circuit partitioned into L gates. Hence, the utilitarian variational quantum programming procedure—based on the classical evaluation of objective functions and iterated feedback—is, in principle, as powerful as any other model of quantum computation. This result elevates the formal standing of the variational approach whereas establishing a universal model of quantum computation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (34)

  1. E. Farhi, J. Goldstone, and S. Gutmann, A quantum approximate optimization algorithm, arXiv:1411.4028.
  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. J. Preskill, Quantum Computing in the NISQ era, and beyond, Quantum 2, 79 (2018).
  4. M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Sci. Technol. 4, 043001 (2019).
  5. S. Hadfield, Z. Wang, B. O'Gorman, E. Rieffel, D. Venturelli, and R. Biswas, From the quantum approximate optimization algorithm to a quantum alternating operator ansatz, Algorithms 12, 34 (2019).
  6. N. Moll, P. Barkoutsos, L. S. Bishop, J. M. Chow, A. Cross, D. J. Egger, S. Filipp, A. Fuhrer, J. M. Gambetta, M. Ganzhorn et al., Quantum optimization using variational algorithms on near-term quantum devices, Quantum Sci. Technol. 3, 030503 (2018).
  7. J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Quantum machine learning, Nature (London) 549, 195 (2017).
  8. G. Verdon, M. Broughton, and J. Biamonte, A quantum algorithm to train neural networks using low-depth circuits, arXiv:1712.05304.
  9. D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev, Adiabatic quantum computation is equivalent to standard quantum computation, in 45th Annual IEEE Symposium on Foundations of Computer Science, Rome, 2004 (IEEE, Piscataway, NJ, 2004), pp. 42–51.
  10. A. M. Childs, Universal Computation by Quantum Walk, Phys. Rev. Lett. 102, 180501 (2009).
  11. N. B. Lovett, S. Cooper, M. Everitt, M. Trevers, and V. Kendon, Universal quantum computation using the discrete-time quantum walk, Phys. Rev. A 81, 042330 (2010).
  12. M. Van den Nest, A. Miyake, W. Dür, and H. J. Briegel, Universal Resources for Measurement-Based Quantum Computation, Phys. Rev. Lett. 97, 150504 (2006).
  13. D. Deutsch, Quantum theory, the Church-Turing principle, and the universal quantum computer, Proc. R. Soc. London A, Ser. A 400, 97 (1985).
  14. S. C. Benjamin and S. Bose, Quantum Computing with An Always-On Heisenberg Interaction, Phys. Rev. Lett. 90, 247901 (2003).
  15. S. C. Benjamin and S. Bose, Quantum computing in arrays coupled by “always-on” interactions, Phys. Rev. A 70, 032314 (2004).
  16. Y. Shi, Both toffoli, and controlled-not need little help to do universal quantum computing, Quantum Info. Comput. 3, 84 (2003).
  17. S. Lloyd, Quantum approximate optimization is computationally universal, arXiv:1812.11075.
  18. M. E. S. Morales, J. D. Biamonte, and Z. Zimborás, On the universality of the quantum approximate optimization algorithm, Quant. Info. Proc. 19, 291 (2020).
  19. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc. 58, 13 (1963).
  20. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevA.103.L030401 for proofs not appearing after a lemma or theorem.
  21. A. Kandala, A. Mezzacapo, K. Temme, M. Takita, M. Brink, J. M. Chow, and J. M. Gambetta, Hardware-efficient variational quantum eigensolver for small molecules, and quantum magnets, Nature (London) 549, 242 (2017).
  22. A. V. Uvarov, A. S. Kardashin, and J. D. Biamonte, Machine learning phase transitions with a quantum processor, Phys. Rev. A 102, 012415 (2020).
  23. A. Kardashin, A. Uvarov, D. Yudin, and J. Biamonte, Certified variational quantum algorithms for eigenstate preparation, Phys. Rev. A 102, 052610 (2020).
  24. L. Cincio, Y. Subaşı, A. T. Sornborger, and P. J. Coles, Learning the quantum algorithm for state overlap, New J. Phys. 20, 113022 (2018).
  25. S. Bravyi, D. Browne, P. Calpin, E. Campbell, D. Gosset, and M. Howard, Simulation of quantum circuits by low-rank stabilizer decompositions, Quantum 3, 181 (2019).
  26. S. Bravyi and D. Gosset, Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates, Phys. Rev. Lett. 116, 250501 (2016).
  27. R. P. Feynman, Quantum mechanical computers, Opt. News 11, 11 (1985).
  28. A. Y. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical, and Quantum Computation (American Mathematical Society, Boston, 2002).
  29. V. Akshay, H. Philathong, M. E. S. Morales, and J. D. Biamonte, Reachability Deficits in Quantum Approximate Optimization, Phys. Rev. Lett. 124, 090504 (2020).
  30. Y. Li and S. C. Benjamin, Efficient Variational Quantum Simulator Incorporating Active Error Minimization, Phys. Rev. X 7, 021050 (2017).
  31. J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New J. Phys. 18, 023023 (2016).
  32. J. D. Biamonte and P. J. Love, Realizable Hamiltonians for universal adiabatic quantum computers, Phys. Rev. A 78, 012352 (2008).
  33. C. Bravo-Prieto, R. LaRose, M. Cerezo, Y. Subasi, L. Cincio, and P. J. Coles, Variational quantum linear solver, arXiv:1909.05820.
  34. M. Lubasch, J. Joo, P. Moinier, M. Kiffner, and D. Jaksch, Variational quantum algorithms for nonlinear problems, Phys. Rev. A 101, 010301(R) (2020).

Outline

Information

Phys. Rev. A 103, L030401– Published 10 March, 2021
  • Received 8 August 2020
  • Revised 29 December 2020
  • Accepted 12 January 2021
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation