Systematic and deterministic graph minor embedding for Cartesian products of graphs

  • Arman Zaribafiyan
  • Dominic J. J. Marchand
  • Seyed Saeed Changiz Rezaei
Article

Abstract

The limited connectivity of current and next-generation quantum annealers motivates the need for efficient graph minor embedding methods. These methods allow non-native problems to be adapted to the target annealer’s architecture. The overhead of the widely used heuristic techniques is quickly proving to be a significant bottleneck for solving real-world applications. To alleviate this difficulty, we propose a systematic and deterministic embedding method, exploiting the structures of both the specific problem and the quantum annealer. We focus on the specific case of the Cartesian product of two complete graphs, a regular structure that occurs in many problems. We decompose the embedding problem by first embedding one of the factors of the Cartesian product in a repeatable pattern. The resulting simplified problem comprises the placement and connecting together of these copies to reach a valid solution. Aside from the obvious advantage of a systematic and deterministic approach with respect to speed and efficiency, the embeddings produced are easily scaled for larger processors and show desirable properties for the number of qubits used and the chain length distribution. We conclude by briefly addressing the problem of circumventing inoperable qubits by presenting possible extensions of our method.

Keywords

Graph minor embedding Cartesian product Quantum annealing Chip architecture 

References

  1. 1.
    Adler, I., Dorn, F., Fomin, F.V., Sau, I., Thilikos, D.M.: Faster parameterized algorithms for minor containment. Theor. Comput. Sci. 412(50), 7018–7028 (2011). doi:10.1016/j.tcs.2011.09.015. http://www.sciencedirect.com/science/article/pii/S0304397511007912
  2. 2.
    Alghassi, H.: The algebraic QUBO design framework. To be publishedGoogle Scholar
  3. 3.
    Bodlaender, H.L., Koster, A.M.: Treewidth computations i. upper bounds. Inf. Comput. 208(3), 259–275 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Boothby, T., King, A.D., Roy, A.: Fast clique minor generation in chimera qubit connectivity graphs. Quantum Inf. Process. 15(1), 495–508 (2016). doi:10.1007/s11128-015-1150-6 ADSMathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors. Preprint (2014). arXiv:1406.2741
  6. 6.
    Choi, V.: Minor-embedding in adiabatic quantum computation: \({\rm II}\). \({\rm M}\)inor-universal graph design. Quantum Inf. Process. 10(3), 343–353 (2011). doi:10.1007/s11128-010-0200-3 MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Dridi, R., Alghassi, H.: Homology computation of large point clouds using quantum annealing. Preprint (2015). arXiv:1512.09328
  8. 8.
    Fan, N., Pardalos, P.M.: Linear and quadratic programming approaches for the general graph partitioning problem. J. Glob. Optim. 48(1), 57–71 (2010). doi:10.1007/s10898-009-9520-1 MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Godsil, C., Royle, G.: Algebraic Graph Theory, Volume 207 of Graduate Texts in Mathematics (2001)Google Scholar
  10. 10.
    Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Comb. Theory Ser. B 99(1), 218–228 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hager, W.W., Krylyuk, Y.: Graph partitioning and continuous quadratic programming. SIAM J. Discret. Math. 12(4), 500–523 (1999)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Hernandez, M., Zaribafiyan, A., Aramon, M., Naghibi, M.: A novel graph-based approach for determining molecular similarity. Preprint (2016). arXiv:1601.06693
  13. 13.
    Imrich, W., Peterin, I.: Recognizing \({\rm C}\)artesian products in linear time. Discret. Math. 307(3–5), 472–483 (2007)MATHCrossRefGoogle Scholar
  14. 14.
    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(7346), 194–198 (2011)ADSCrossRefGoogle Scholar
  15. 15.
    Kaminsky, W., Lloyd, S.: Scalable architecture for adiabatic quantum computing of \(\rm NP\)-hard problems. In: Leggett, A., Ruggiero, B., Silvestrini, P. (eds.) Quantum Computing and Quantum Bits in Mesoscopic Systems, pp. 229–236. Springer, New York (2004). doi:10.1007/978-1-4419-9092-1_25
  16. 16.
    Kaplansky, I., Riordan, J.: The problem of the rooks and its applications. Duke Math. J. 13(2), 259–268 (1946). doi:10.1215/S0012-7094-46-01324-5 MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    King, A.D., Lanting, T., Harris, R.: Performance of a quantum annealer on range-limited constraint satisfaction problems. Preprint (2015). arXiv:1502.02098
  18. 18.
    Lucena, B.: Achievable sets, brambles, and sparse treewidth obstructions. Discret. Appl. Math. 155(8), 1055–1065 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Pardalos, P.M., Mavridou, T., Xue, J.: Handbook of Combinatorial Optimization: volume 2, chap. The Graph Coloring Problem: A Bibliographic Survey, pp. 1077–1141. Springer, Boston (1999). doi:10.1007/978-1-4613-0303-9_16
  20. 20.
    Perdomo-Ortiz, A., Fluegemann, J., Biswas, R., Smelyanskiy, V.N.: A performance estimator for quantum annealers: gauge selection and parameter setting. Preprint (2015). arXiv:1503.01083
  21. 21.
    Perdomo-Ortiz, A., O’Gorman, B., Fluegemann, J., Biswas, R., Smelyanskiy, V.N.: Determination and correction of persistent biases in quantum annealers. Sci. Rep. 6, 18628 (2016)Google Scholar
  22. 22.
    Rieffel, E.G., Venturelli, D., O’Gorman, B., Do, M.B., Prystay, E.M., Smelyanskiy, V.N.: A case study in programming a quantum annealer for hard operational planning problems. Quantum Inf. Process. 14(1), 1–36 (2014). doi:10.1007/s11128-014-0892-x ADSMATHCrossRefGoogle Scholar
  23. 23.
    Rieffel, E.G., Venturelli, D., O’Gorman, B., Do, M.B., Prystay, E.M., Smelyanskiy, V.N.: A case study in programming a quantum annealer for hard operational planning problems. Quantum Inf. Process. 14(1), 1–36 (2015)ADSMATHCrossRefGoogle Scholar
  24. 24.
    Seymour, P.D., Thomas, R.: Graph searching and a min–max theorem for tree-width. J. Comb. Theory Ser. B 58(1), 22–33 (1993)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Thomas, R.: Tree-decompositions of graphs (lecture notes). School of Mathematics. Georgia Institute of Technology, Atlanta, 30332 (1996)Google Scholar
  26. 26.
    Venturelli, D., Mandrà, S., Knysh, S., O’Gorman, B., Biswas, R., Smelyanskiy, V.: Quantum optimization of fully connected spin glasses. Phys. Rev. X 5, 031040 (2015). doi:10.1103/PhysRevX.5.031040 Google Scholar
  27. 27.
    Venturelli, D., Marchand, D.J.J., Rojo, G.: Quantum annealing implementation of job-shop scheduling. Preprint (2015). arXiv:1506.08479
  28. 28.
    Young, K.C., Blume-Kohout, R., Lidar, D.A.: Adiabatic quantum optimization with the wrong Hamiltonian. Phys. Rev. A 88, 062314 (2013). doi:10.1103/PhysRevA.88.062314 ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Arman Zaribafiyan
    • 1
  • Dominic J. J. Marchand
    • 1
  • Seyed Saeed Changiz Rezaei
    • 1
  1. 1.1QB Information Technologies (1QBit)VancouverCanada

Personalised recommendations