Skip to main content
SpringerLink
Log in

The effects of the problem Hamiltonian parameters on the minimum spectral gap in adiabatic quantum optimization

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We study the relation between the Ising problem Hamiltonian parameters and the minimum spectral gap (min-gap) of the system Hamiltonian in the Ising-based quantum annealer. The main criterion we use in this paper to assess the performance of a QA algorithm is the presence or absence of an anti-crossing during quantum evolution. For this purpose, we introduce a new parametrization definition of the anti-crossing. Using the Maximum-weighted Independent Set (MIS) problem in which there are flexible parameters (energy penalties J between pairs of edges) in an Ising formulation as the model problem, we construct examples to show that by changing the value of J, we can change the quantum evolution from one that has an anti-crossing (that results in an exponential small min-gap) to one that does not have, or the other way around, and thus drastically change (increase or decrease) the min-gap. However, we also show that by changing the value of J alone, one can not avoid the anti-crossing. We recall a polynomial reduction from an Ising problem to an MIS problem to show that the flexibility of changing parameters without changing the problem to be solved can be applied to any Ising problem. As an example, we show that by such a reduction alone, it is possible to remove the anti-crossing and thus increase the min-gap. Our anti-crossing definition is necessarily scaling invariant as scaling the problem Hamiltonian does not change the nature (i.e., presence or absence) of an anti-crossing. As a side note, we show exactly how the min-gap is scaled if we scale the problem Hamiltonian by a constant factor.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Notes

  1. We purposely left out the results for \(J=2\) as an exercise for the curious reader.

  2. In [16], an anti-crossing is identified by “the local minimum in the plot of the instantaneous energy gap as a function of time”.

  3. See [7] for a correction to [4].

References

  1. Albash, T.: Role of non-stoquastic catalysts in quantum adiabatic optimization. Phys. Rev. A 99, 042334 (2019)

    Article  ADS  Google Scholar 

  2. Albash, T.: Private Communication (2019)

  3. Albash, T., Lidar, D.A.: Adiabatic quantum computation. Rev. Mod. Phys. 90, 015002 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  4. Altshuler, B., Krovi, H., Roland, J.: Anderson localization makes adiabatic quantum optimization fail. Proc. Natl. Acad. Sci. USA 107, 12446–12450 (2010)

    Article  ADS  Google Scholar 

  5. Amin, M.H.S., Choi, V.: First order phase transition in adiabatic quantum computation. Phys. Rev. A 80(6), (2009) arXiv:0904.1387 [quant-ph]

  6. Boros, E., Hammer, P.L.: Pseudo-Boolean optimization. Discret. Appl. Math. 123, 155–225 (2002)

    Article  MathSciNet  Google Scholar 

  7. Choi, V.: Different adiabatic quantum algorithms for the NP-complete exact cover problem. Proc. Natl. Acad. Sci. USA 108(7), E19–E20 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  8. Choi, V.: Different adiabatic quantum algorithms for the NP-complete exact cover and 3SAT problem. Quantum Inf. Comput. 11, 0638–0648 (2011). arXiv:1010.1221 [quant-ph]

    MathSciNet  Google Scholar 

  9. Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7, 193–209 (2008)

    Article  MathSciNet  Google Scholar 

  10. Choi, V.: XX-Driver Hamiltonian and driver architecture in adiabatic quantum optimization. In: preparation (2019)

  11. Dickson, N.G., Amin, M.H.S.: Does adiabatic quantum optimization fail for NP-complete problems? Phys. Rev. Lett. 106, 050502 (2011)

    Article  ADS  Google Scholar 

  12. Dickson, N.G.: Elimination of perturbative crossings in adiabatic quantum optimization. New J. Phys. 13, 073011 (2011)

    Article  ADS  Google Scholar 

  13. Farhi, E., Gosset, D., Hen, I., Sandvik, A.W., Shor, P., Young, A.P., Zamponi, F.: The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs. Phys. Rev. A 86, 052334 (2012)

    Article  ADS  Google Scholar 

  14. Fujii, K.: Quantum speedup in stoquastic adiabatic quantum computation. arXiv:1803.09954 (2018)

  15. Hastings, M.B., Freedman, M.H.: Obstructions to classically simulating the quantum adiabatic algorithm. arXiv:1302.5733 (2013)

  16. Hormozi, L., Brown, E.W., Carleo, G., Troyer, M.: Nonstoquastic Hamiltonians and quantum annealing of an ising spin glass. Phys. Rev. B 95, 184416 (2017)

    Article  ADS  Google Scholar 

  17. Jarret, M., Jordan, S.P., Lackey, B.: Adiabatic optimization versus diffusion Monte Carlo methods. Phys. Rev. A 94, 042318 (2016)

    Article  ADS  Google Scholar 

  18. Knysh, S.: Zero-temperature quantum annealing bottlenecks in the spin-glass phase. Nat. Commun. 7, 12370 (2016)

    Article  ADS  Google Scholar 

  19. Muthukrishnan, S., Albash, T., Lidar, D.A.: Tunneling and speedup in quantum optimization for permutation-symmetric problems. Phys. Rev. X 6, 031010 (2016) arXiv:1511.03910 [quant-ph]

  20. Qureshi, M.A., Zhong, J., Mason, P., Betouras, J.J., Zagoskin, A. M.: Pechukas–Yukawa formalism for Landau–Zener transitions in the presence of external noise (2018) arXiv:1803.05034v2

  21. Stein, W.A. et al.: Sage mathematics software (Version 8.0). The Sage Development Team (2009). http://www.sagemath.org

  22. Wilkinson, M.: Statistical aspects of dissipation by Landau–Zener transitions. J. Phys. A Math. Gen. 21, 4021–4037 (1988)

    Article  ADS  Google Scholar 

  23. Wilkinson, M.: Statistics of multiple avoided crossings. J. Phys. A Math. Gen. 22, 2795–2805 (1989)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author is grateful to Tameem Albash for his generous sharing/update knowledge of the current state of art the QA algorithm, especially in regard to topics including the min-gap estimation, anti-crossing and perturbative crossing, quantum tunneling; and for his many comments on this paper; and for the permission to use the loop gadgets example (including Fig. 2) he constructed in this paper. Special thanks also go to Daniel Lidar, for the invitation to resume the AQC research, and for his comments on this paper. I would also like to thank Adrian Lupascu and his QEO group members for the comments. The author also gratefully acknowledges the use of the Sage Mathematics Software [21] for the computation of the eigenvalues and plotting the graphs in this paper. The research is based upon work partially supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office contract W911NF-17-C-0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.

Author information

Authors and Affiliations

  1. Gladiolus Veritatis Consulting Co., 33768 Cherry Ave, Mission, BC, V2V2V6, Canada

    Vicky Choi

Authors
  1. Vicky Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vicky Choi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Choi, V. The effects of the problem Hamiltonian parameters on the minimum spectral gap in adiabatic quantum optimization. Quantum Inf Process 19, 90 (2020). https://doi.org/10.1007/s11128-020-2582-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-020-2582-1

Keywords

Search

Navigation