Advertisement

Quantum Information Processing

, Volume 7, Issue 5, pp 193–209 | Cite as

Minor-embedding in adiabatic quantum computation: I. The parameter setting problem

  • Vicky ChoiEmail author
Article

Abstract

We show that the NP-hard quadratic unconstrained binary optimization (QUBO) problem on a graph G can be solved using an adiabatic quantum computer that implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding of G in the quantum hardware graph U. There are two components to this reduction: embedding and parameter setting. The embedding problem is to find a minor-embedding G emb of a graph G in U, which is a subgraph of U such that G can be obtained from G emb by contracting edges. The parameter setting problem is to determine the corresponding parameters, qubit biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper, we focus on the parameter setting problem. As an example, we demonstrate the embedded Ising Hamiltonian for solving the maximum independent set (MIS) problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system. We close by discussing several related algorithmic problems that need to be investigated in order to facilitate the design of adiabatic algorithms and AQC architectures.

Keywords

Adiabatic quantum computation Graph minor Quadratic unconstrained binary optimization Ising Hamiltonian 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev,O.: Adiabatic quantum computation is equaivalent to standard quantum computation. In: Proc. 45th FOCS, pp. 42–51 (2004)Google Scholar
  2. 2.
    Amin M.H.S., Love P.J., Truncik C.J.S.: Thermally assisted adiabatic quantum computation. Phys. Rev. Lett. 100, 060503 (2008)CrossRefADSGoogle Scholar
  3. 3.
    Amin, M.H.S., Truncik, C.J.S., Averin, D.V.: The role of single qubit decoherence time in adiabatic quantum computation. arXiv:quant-ph/0803.1196 (2008)Google Scholar
  4. 4.
    Baker B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bansal, N., Bravyi, S., Terhal, B.M.: A classical approximation scheme for the ground-state energy of Ising spin Hamiltonians on planar graphs. arXiv:quant-ph/0705.1115 (2007)Google Scholar
  6. 6.
    Barahona F.: On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15, 3241–3253 (1982)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Boros E., Hammer P.: Pseudo-Boolean optimization. Discrete Appl. Math. 123, 155–225 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of quadratic unconstrained binary optimization. Technical Report RRR 10-2006, RUTCOR Research Report (2006)Google Scholar
  9. 9.
    Bravyi, S., DiVincenzo, D.P., Loss, D., Terhal, B.M.: Simulation of many-body Hamiltonians using perturbation theory with bounded-strength interactions. arXiv:quant-ph/0803.2686 (2008)Google Scholar
  10. 10.
    Childs, A., Farhi, E., Preskill, J.: Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (10 pp.) (2001)Google Scholar
  11. 11.
    Diestel R.: Graph Theory. Springer-Verlag, Heidelberg (2005)zbMATHGoogle Scholar
  12. 12.
    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(5516), 472–476 (2001)CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. arXiv:quant-ph/0001106 (2000)Google Scholar
  14. 14.
    Ioannou, L.M., Mosca, M.: Limitations of some simple adiabatic quantum algorithms. arXiv:quant-ph/ 0702241 (2007)Google Scholar
  15. 15.
    Kaminsky W.M., Lloyd S.: Scalable architecture for adiabatic quantum computing of NP-hard problems. In: Leggett, A.J., Ruggiero, B., Silvestrini, P. (eds) Quantum Computing and Quantum Bits in Mesoscopic Systems, Kluwer, New York (2004)Google Scholar
  16. 16.
    Kaminsky, W.M., Lloyd, S., Orlando, T.P.: Scalable superconducting architecture for adiabatic quantum computation. arXiv:quant-ph/0403090 (2004)Google Scholar
  17. 17.
    Kempe J., Kitaev A., Regev., O: The complexity of the local Hamiltonian problem. SIAM J. Comput. 35, 1070 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kleinberg, J.M., Rubinfeld, R.: Short paths in expander graphs. In: IEEE Symposium on Foundations of Computer Science, pp. 86–95 (1996)Google Scholar
  19. 19.
    Lidar, D.A., Rezakhani, A.T., Hamma, A.: Adiabatic approximation with better than exponential accuracy for many-body systems and quantum computation. arXiv:quant-ph/0808.2697 (2008)Google Scholar
  20. 20.
    Oliveira, R., Terhal, B.M.: The complexity of quantum spin systems on a two-dimensional square lattice. arXiv:quant-ph/0504050 (2005)Google Scholar
  21. 21.
    Reichardt, B.W.: The quantum adiabatic optimization algorithm and local minima. In: STOC ’04: Proceedings of the Thirty-sixth Annual ACM Symposium on Theory of Computing, pp. 502–510. ACM, New York, NY, USA (2004)Google Scholar
  22. 22.
    Robertson N., Seymour P.D.: Graph minors. xiii: the disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Thomson, L.F.: Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes. Morgan Kaufmann, San Mateo, California (1992)Google Scholar
  24. 24.
    van Dam, W., Mosca, M., Vazirani, U.: How powerful is adiabatic quantum computation? In: Proc. 42nd FOCS, pp. 279–287 (2001)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.D-Wave Systems Inc.BurnabyCanada

Personalised recommendations