Skip to main content
SpringerLink
Log in

How quantum is the speedup in adiabatic unstructured search?

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In classical computing, analog approaches have sometimes appeared to be more powerful than they really are. This occurs when resources, particularly precision, are not appropriately taken into account. While the same should also hold for analog quantum computing, precision issues are often neglected from the analysis. In this work, we present a classical analog algorithm for unstructured search that can be viewed as analogous to the quantum adiabatic unstructured search algorithm devised by Roland and Cerf (Phys Rev A 65:042308, 2002). We show that similarly to its quantum counterpart, the classical construction may also provide a quadratic speedup over standard digital unstructured search. We discuss the meaning and the possible implications of this result in the context of adiabatic quantum computing.

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

Notes

  1. The Lagrangian can be interpreted as a system of rotors with time-varying potential and moments of inertia.

  2. One could consider a small addition to the kinetic term that would remove the degeneracy of its minima and enforce the \(\theta _i=\pi /2\) condition.

  3. In this case, the algorithm is reminiscent of the analog quantum search algorithm proposed in Ref. [24].

References

  1. Shor, P.W.: Algorithms for quantum computing: discrete logarithms and factoring. In: Goldwasser, S. (ed.) Proceedings of the 35th Symposium on Foundations of Computer Science, p. 124 (1994)

  2. Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Phys. Rev. Lett. 79, 325 (1997)

    Article  ADS  Google Scholar 

  3. Finnila, A.B., Gomez, M.A., Sebenik, C., Stenson, C., Doll, J.D.: Quantum annealing: a new method for minimizing multidimensional functions. Chem. Phys. Lett. 219, 343–348 (1994)

    Article  ADS  Google Scholar 

  4. Brooke, J., Bitko, D., Rosenbaum, T.F., Aeppli, G.: Quantum annealing of a disordered magnet. Science 284, 779–781 (1999)

    Article  ADS  Google Scholar 

  5. Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58, 5355 (1998)

    Article  ADS  Google Scholar 

  6. Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292, 472 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  7. Santoro, G., Martoňák, R., Tosatti, E., Car, R.: Theory of quantum annealing of an Ising spin glass. Science 295, 2427 (2002)

    Article  ADS  Google Scholar 

  8. Vandersypen, L.M.K., Steffen, M., Breyta, G., Yannoni, C.S., Sherwood, M.H., Chuang, I.L.: Experimental realization of shors quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883887 (2001)

    Article  Google Scholar 

  9. Bian, Z., Chudak, F., Macready, W.G., Clark, L., Gaitan, F.: Experimental determination of Ramsey numbers with quantum annealing (2012). arXiv:1201.1842

  10. Rønnow, T.F., Wang, Z., Job, J., Boixo, S., Isakov, S.V., Wecker, D., Martinis, J.M., Lidar, D.A., Troyer, M.: Defining and detecting quantum speedup. Science 345, 420–424 (2014)

    Article  ADS  Google Scholar 

  11. Johnson, M.W., Amin, M.H.S., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Harris, R., Berkley, A.J., Johansson, J., Bunyk, P., Chapple, E.M., Enderud, C., Hilton, J.P., Karimi, K., Ladizinsky, E., Ladizinsky, N., Oh, T., Perminov, I., Rich, C., Thom, M.C., Tolkacheva, E., Truncik, C.J.S., Uchaikin, S., Wang, J., Wilson, B., Rose, G.: Quantum annealing with manufactured spins. Nature 473, 194–198 (2011)

    Article  ADS  Google Scholar 

  12. Hen, I., Job, J., Albash, T., Rønnow, T.F., Troyer, M., Lidar, D.A.: Probing for quantum speedup in spinglass problems with planted solutions. Phys. Rev. A 92, 042325 (2015)

    Article  ADS  Google Scholar 

  13. Boixo, S., Albash, T., Spedalieri, F.M., Chancellor, N., Lidar, D.A.: Experimental signature of programmable quantum annealing. Nat. Commun. 4, 2067 (2013)

    Article  ADS  Google Scholar 

  14. Albash, T., Vinci, W., Mishra, A., Warburton, P.A., Lidar, D.A.: Consistency tests of classical and quantum models for a quantum annealer. Phys. Rev. A 91, 042314 (2015)

    Article  ADS  Google Scholar 

  15. Tolpygo, S.K., Bolkhovsky, V., Weir, T.J., Johnson, L.M., Gouker, M.A., Oliver, W.D.: Fabrication process and properties of fully-planarized deep-submicron Nb/Al-AlOx/Nb Josephson junctions for VLSI circuits. IEEE Trans. Appl. Supercond. 25, 1–12 (2015)

    Google Scholar 

  16. Tolpygo, S.K., Bolkhovsky, V., Weir, T.J., Galbraith, C.J., Johnson, L.M., Gouker, M.A., Semenov, V.K.: Inductance of circuit structures for MIT LL superconductor electronics fabrication process with 8 niobium layers. IEEE Trans. Appl. Supercond. 25, 1–5 (2015)

    Google Scholar 

  17. Jin, X.Y., Kamal, A., Sears, A.P., Gudmundsen, T., Hover, D., Miloshi, J., Slattery, R., Yan, F., Yoder, J., Orlando, T.P., Gustavsson, S., Oliver, W.D.: Thermal and residual excited-state population in a 3d transmon qubit. Phys. Rev. Lett. 114, 240501 (2015)

    Article  ADS  Google Scholar 

  18. Young, A.P., Knysh, S., Smelyanskiy, V.N.: Size dependence of the minimum excitation gap in the Quantum Adiabatic Algorithm. Phys. Rev. Lett. 101, 170503 (2008). arXiv:0803.3971

    Article  ADS  Google Scholar 

  19. Young, A.P., Knysh, S., Smelyanskiy, V.N.: First order phase transition in the Quantum Adiabatic Algorithm. Phys. Rev. Lett. 104, 020502 (2010). arXiv:0910.1378

    Article  ADS  Google Scholar 

  20. Hen, I., Young, A.P.: Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems. Phys. Rev. E. 84, 061152 (2011). arXiv:1109.6872v2

    Article  ADS  Google Scholar 

  21. Hen, I.: Excitation gap from optimized correlation functions in quantum Monte Carlo simulations. Phys. Rev. E. 85, 036705 (2012). arXiv:1112.2269v2

    Article  ADS  Google Scholar 

  22. Farhi, E., Gosset, D., Hen, I., Sandvik, A.W., Shor, P., Young, A.P., Zamponi, F.: 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 

  23. Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)

    Article  ADS  Google Scholar 

  24. Farhi, E., Gutmann, S.: An Analog Analogue of a Digital Quantum Computation. arXiv:quant-ph/9612026 (1996)

  25. van Dam, W., Mosca, M., Vazirani, U.: How powerful is adiabatic quantum computation?. In: Proceedings 2001 IEEE International Conference on Cluster Computing, pp. 279–287 (2001)

  26. Jansen, S., Ruskai, M.B., Seiler, R.: Bounds for the adiabatic approximation with applications to quantum computation. J. Math. Phys. 47, 102111 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  27. Lidar, D.A., Rezakhani, A.T., Hamma, A.: Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J. Math. Phys. 50, 102106 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  28. Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn. 5, 435 (1951)

    Article  ADS  Google Scholar 

  29. Hen, I.: Continuous-time quantum algorithms for unstructured problems. J. Phys. A Math. Theor. 47, 045305 (2014). arXiv:1302.7256

    Article  ADS  MathSciNet  Google Scholar 

  30. Hen, I.: How fast can quantum annealers count? J. Phys. A Math. Theor. 47, 235304 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  31. Hen, I.: Realizable quantum adiabatic search. EPL (Europhys. Lett.) 118, 30003 (2017)

    Article  ADS  Google Scholar 

  32. Roland, J., Cerf, N.J.: Adiabatic quantum search algorithm for structured problems. Phys. Rev. A 68, 062312 (2003)

    Article  ADS  Google Scholar 

  33. Andrew, M.: Childs and Jeffrey Goldstone, Spatial search by quantum walk. Phys. Rev. A 70, 022314 (2004)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  35. Jiang, Z., Rieffel, E.G., Wang, Z.: Near-optimal quantum circuit for grover’s unstructured search using a transverse field. Phys. Rev. A 95, 062317 (2017)

    Article  ADS  Google Scholar 

  36. Aaronson, S.: Guest column: Np-complete problems and physical reality. SIGACT News 36, 30–52 (2005)

    Article  Google Scholar 

  37. Vergis, A., Steiglitz, K., Dickinson, B.: The complexity of analog computation. Math. Comput. Simul. 28, 91–113 (1986)

    Article  Google Scholar 

  38. Albert, S.: Jackson, Analog Computation. McGraw-Hill, New York (1960)

    Google Scholar 

  39. Albash, T., Martin-Mayor, V., Hen, I.: Analog errors in ising machines. Accepted for publication in Quantum Science and Technology (2019). arXiv:1806.03744 [quant-ph]

  40. Jonckheere, E.A., Rezakhani, A.T., Ahmad, F.: Differential topology of adiabatically controlled quantum processes. Quantum Inf. Process. 12, 1515–1538 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  41. Slutskii, M., Albash, T., Barash, L., Hen, I.: Analog nature of quantum adiabatic unstructured search. (2019, in preparation). arXiv:1904.04420

  42. Berry, D.W., Cleve, R., Gharibian, S.: Gate-efficient discrete simulations of continuous-time quantum query algorithms. Quantum Info. Comput. 14, 1–30 (2014)

    MathSciNet  Google Scholar 

  43. Berry, D.W., Childs, A.M., Cleve, R., Kothari, R., Somma, R.D.: Exponential improvement in precision for simulating sparse hamiltonians. In: Proceedings of the Forty-sixth Annual ACM Symposium on Theory of Computing, STOC ’14 (ACM, New York, NY, USA) pp. 283–292 (2014)

  44. Berry, D.W., Childs, A.M., Cleve, R., Kothari, R., Somma, R.D.: Simulating hamiltonian dynamics with a truncated taylor series. Phys. Rev. Lett. 114, 090502 (2015)

    Article  ADS  Google Scholar 

  45. Cleve, R., Gottesman, D., Mosca, M., Somma, R.D., Yonge-Mallo, D.: Efficient discrete-time simulations of continuous-time quantum query algorithms. In: Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing, STOC ’09 (ACM, New York, NY, USA) pp. 409–416 (2009)

  46. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

Download references

Acknowledgements

We thank Tameem Albash, Elizabeth Crosson, Daniel Lidar and Eleanor Rieffel for insightful discussions. 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. Information Sciences Institute, University of Southern California, Marina del Rey, CA, 90292, USA

    Itay Hen

  2. Department of Physics and Astronomy, Center for Quantum Information Science and Technology, University of Southern California, Los Angeles, CA, 90089, USA

    Itay Hen

Authors
  1. Itay Hen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Itay Hen.

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

Hen, I. How quantum is the speedup in adiabatic unstructured search?. Quantum Inf Process 18, 162 (2019). https://doi.org/10.1007/s11128-019-2281-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2281-y

Keywords

Search

Navigation