Quantum Information Processing

, Volume 14, Issue 12, pp 4731–4749 | Cite as

Challenges of adiabatic quantum evaluation of NAND trees



The quantum adiabatic algorithm solves problems by evolving a known initial state towards the ground state of a Hamiltonian encoding the answer to a problem. Although continuous- and discrete-time quantum algorithms exist capable of evaluating tree graphs consisting of N vertexes in \(O(\sqrt{N})\) time, a quadratic improvement over their classical counterparts, no quantum adiabatic procedure is known to exist. In this work, we present a study of the main issues and challenges surrounding quantum adiabatic evaluation of NAND trees. We focus on a number of issues ranging from: (1) mapping mechanisms; (2) spectrum analysis and remapping; (3) numerical evaluation of spectrum gaps; and (4) algorithmic procedures. These concepts are then used to provide numerical evidence for the existence of a \(\frac{N^{2}}{\log {N^{2}}}\) gap.


Quantum adiabatic computation Tree evaluation Spectrum evaluation Numerical analysis 

Mathematics Subject Classification

81P68 68Q05 68Q10 68Q12 


  1. 1.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. eprint arXiv:quant-ph/0405098 (2004)
  2. 2.
    Altshuler, B., Krovi, H., Roland, J.: Adiabatic quantum optimization fails for random instances of NP-complete problems. ArXiv e-prints (2009)Google Scholar
  3. 3.
    Ambainis, A.: A nearly optimal discrete query quantum algorithm for evaluating NAND formulas. ArXiv e-prints (2007)Google Scholar
  4. 4.
    Ambainis, A., Childs, A.M., Reichardt, B.W., Spalek, R., Zhang, S.: Any AND-OR formula of size N can be evaluated in time \(N^{\frac{1}{2} + o(1)}\) on a quantum computer. In: Proceedings of the 48th annual IEEE symposium on foundations of computer science, FOCS ’07, pp. 363 –372. IEEE Computer Society, Washington, DC, USA (2007). doi:10.1109/FOCS.2007.10
  5. 5.
    Avron, J.E., Elgart, A.: Adiabatic theorem without a gap condition. Commun. Math. Phys. 203(2), 445–463 (1999). doi:10.1007/s002200050620 MathSciNetCrossRefADSMATHGoogle Scholar
  6. 6.
    Bennett, C.H., Bernstein, E., Brassard, G., Vazirani, U.: Strengths and weaknesses of quantum computing. eprint arXiv:quant-ph/9701001 (1997)
  7. 7.
    Biggs, N.: Algebraic Graph Theory. Cambridge Mathematical Library. Cambridge University Press (1993). http://books.google.com.br/books?id=6TasRmIFOxQC
  8. 8.
    Born, M., Fock, V.: Beweis des adiabatensatzes. Zeitschrift für Physik 51(3–4), 165–180 (1928). doi:10.1007/BF01343193 CrossRefADSMATHGoogle Scholar
  9. 9.
    Cerf, N.J., Grover, L.K., Williams, C.P.: Nested quantum search and structured problems. Phys. Rev. A 61(3), 032,303 (2000). doi:10.1103/PhysRevA.61.032303 CrossRefGoogle Scholar
  10. 10.
    Childs, A.M., Cleve, R., Jordan, S.P., Yonge-Mallo, D.: Discrete-query quantum algorithm for NAND trees. Theory Comput. 5(1), 119–123 (2009). doi:10.4086/toc.2009.v005a005 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Childs, A.M., Reichardt, B.W., Spalek, R., Zhang, S.: Every NAND formula of size N can be evaluated in time \(N^{\frac{1}{2}+o(1)}\) on a quantum computer. eprint arXiv:quant-ph/0703015 (2007)
  12. 12.
    Cleve, R., Gavinsky, D., Yonge-Mallo, D.: Quantum algorithms for evaluating min-max trees. In: Kawano, Y., Mosca, M. (eds.) Proceedings of theory of quantum computation, communication, and cryptography (TQC 2008), vol. 5106, pp. 11–15. Springer, Berlin (2008)Google Scholar
  13. 13.
    Farhi, E., Goldstone, J., Gosset, D., Gutmann, S., Meyer, H.B., Shor, P.: Quantum adiabatic algorithms, small gaps, and different paths (2009). http://www.citebase.org/abstract?id=oai:arXiv.org:0909.4766
  14. 14.
    Farhi, E., Goldstone, J., Gutmann, S.: A numerical study of the performance of a quantum adiabatic evolution algorithm for satisfiability. eprint arXiv:quant-ph/0007071 (2000)
  15. 15.
    Farhi, E., Goldstone, J., Gutmann, S.: A quantum algorithm for the Hamiltonian NAND tree. Theory Comput. 4(1), 169–190 (2008). doi:10.4086/toc.2008.v004a008 MathSciNetCrossRefGoogle Scholar
  16. 16.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. ArXiv Quantum Physics e-prints (2000)Google Scholar
  17. 17.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: STOC ’96: Proceedings of the twenty-eighth annual ACM symposium on theory of computing, pp. 212–219. ACM, New York, NY, USA (1996). doi: 10.1145/237814.237866
  18. 18.
    Mochon, C.: Hamiltonian oracles. Phys. Rev. A 75, 042,313 (2007). doi:10.1103/PhysRevA.75.042313 CrossRefGoogle Scholar
  19. 19.
    Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach. Prentice Hall Series in Artificial Intelligence. Prentice Hall, Englewood Cliffs (2010)Google Scholar
  20. 20.
    Saks, M., Wigderson, A.: Probabilistic Boolean decision trees and the complexity of evaluating game trees. In: Proceedings of the 27th annual symposium on foundations of computer science, SFCS ’86, pp. 29–38. IEEE Computer Society, Washington, DC, USA (1986). doi:10.1109/SFCS.1986.44
  21. 21.
    van Dam, W., Mosca, M., Vazirani, U.: How powerful is adiabatic quantum computation? In: 42nd IEEE symposium on foundations of computer science, 2001. Proceedings, pp. 279–287 (2001). doi:10.1109/SFCS.2001.959902

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.QCG/LNCCPetrópolisBrazil

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