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Analyzing the Quantum Annealing Approach for Solving Linear Least Squares Problems

  • Ajinkya BorleEmail author
  • Samuel J. Lomonaco
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

Abstract

With the advent of quantum computers, researchers are exploring if quantum mechanics can be leveraged to solve important problems in ways that may provide advantages not possible with conventional or classical methods. A previous work by O’Malley and Vesselinov in 2016 briefly explored using a quantum annealing machine for solving linear least squares problems for real numbers. They suggested that it is best suited for binary and sparse versions of the problem. In our work, we propose a more compact way to represent variables using two’s and one’s complement on a quantum annealer. We then do an in-depth theoretical analysis of this approach, showing the conditions for which this method may be able to outperform the traditional classical methods for solving general linear least squares problems. Finally, based on our analysis and observations, we discuss potentially promising areas of further research where quantum annealing can be especially beneficial.

Keywords

Quantum annealing Simulated annealing Quantum computing Combinatorial optimization Linear least squares Numerical methods 

Notes

Acknowledgement

The authors would like to thank Daniel O’Malley from LANL for his feedback. A special thanks to John Dorband, whose suggestions inspired the development the one’s/two’s complement encoding. Finally, the authors would like to thank Milton Halem of UMBC and D-wave Systems for providing access to their machines.

References

  1. 1.
    Quantum enhanced optimization (qeo). https://www.iarpa.gov/index.php/research-programs/qeo
  2. 2.
    Adachi, S.H., Henderson, M.P.: Application of quantum annealing to training of deep neural networks. arXiv preprint arXiv:1510.06356 (2015)
  3. 3.
    Aramon, M., Rosenberg, G., Miyazawa, T., Tamura, H., Katzgraber, H.G.: Physics-inspired optimization for constraint-satisfaction problems using a digital annealer. arXiv preprint arXiv:1806.08815 (2018)
  4. 4.
    Boixo, S., et al.: Evidence for quantum annealing with more than one hundred qubits. Nat. Phys. 10(3), 218 (2014)CrossRefGoogle Scholar
  5. 5.
    Chang, X.W., Han, Q.: Solving box-constrained integer least squares problems. IEEE Trans. Wirel. Commun. 7(1), 277–287 (2008)CrossRefGoogle Scholar
  6. 6.
    Do, Q.L.: Numerically efficient methods for solving least squares problems (2012)Google Scholar
  7. 7.
    Dorband, J.E.: Stochastic characteristics of qubits and qubit chains on the D-wave 2X. arXiv preprint arXiv:1606.05550 (2016)
  8. 8.
    Dorband, J.E.: A method of finding a lower energy solution to a QUBO/Ising objective function. arXiv preprint arXiv:1801.04849 (2018)
  9. 9.
    Drineas, P., Mahoney, M.W., Muthukrishnan, S., Sarlós, T.: Faster least squares approximation. Numer. math. 117(2), 219–249 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Grote, M.J., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18(3), 838–853 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Honjo, T., Inagaki, T., Inaba, K., Ikuta, T., Takesue, H.: Long-term stable operation of coherent Ising machine for cloud service. In: CLEO: Science and Innovations, pp. JTu2A-87. Optical Society of America (2018)Google Scholar
  12. 12.
    Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58(5), 5355 (1998)CrossRefGoogle Scholar
  13. 13.
    Karimi, K., et al.: Investigating the performance of an adiabatic quantum optimization processor. Quantum Inf. Process. 11(1), 77–88 (2012)CrossRefGoogle Scholar
  14. 14.
    OGorman, B., Babbush, R., Perdomo-Ortiz, A., Aspuru-Guzik, A., Smelyanskiy, V.: Bayesian network structure learning using quantum annealing. Eur. Phys. J. Spec. Top. 224(1), 163–188 (2015)CrossRefGoogle Scholar
  15. 15.
    O’Malley, D., Vesselinov, V.V.: ToQ. jl: A high-level programming language for D-wave machines based on Julia. In: 2016 IEEE High Performance Extreme Computing Conference (HPEC), pp. 1–7. IEEE (2016)Google Scholar
  16. 16.
    O’Malley, D., Vesselinov, V.V., Alexandrov, B.S., Alexandrov, L.B.: Nonnegative/binary matrix factorization with a D-wave quantum annealer. arXiv preprint arXiv:1704.01605 (2017)
  17. 17.
    Pilanci, M., Wainwright, M.J.: Iterative Hessian sketch: fast and accurate solution approximation for constrained least-squares. J. Mach. Learn. Res. 17(1), 1842–1879 (2016)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Tanaka, S., Tamura, R., Chakrabarti, B.K.: Quantum Spin Glasses, Annealing and Computation. Cambridge University Press, Cambridge (2017)zbMATHGoogle Scholar
  19. 19.
    Tsakonas, E., Jaldén, J., Ottersten, B.: Robust binary least squares: relaxations and algorithms. In: 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3780–3783. IEEE (2011)Google Scholar
  20. 20.
    Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wang, G.: Quantum algorithm for linear regression. Phys. Rev. A 96(1), 012335 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.CSEE DepartmentUniversity of Maryland Baltimore CountyBaltimoreUSA

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