Computational Optimization and Applications

, Volume 65, Issue 3, pp 845–869 | Cite as

Building an iterative heuristic solver for a quantum annealer

  • Gili RosenbergEmail author
  • Mohammad Vazifeh
  • Brad Woods
  • Eldad Haber


A quantum annealer heuristically minimizes quadratic unconstrained binary optimization (QUBO) problems, but is limited by the physical hardware in the size and density of the problems it can handle. We have developed a meta-heuristic solver that utilizes D-Wave Systems’ quantum annealer (or any other QUBO problem optimizer) to solve larger or denser problems, by iteratively solving subproblems, while keeping the rest of the variables fixed. We present our algorithm, several variants, and the results for the optimization of standard QUBO problem instances from OR-Library of sizes 500 and 2500 as well as the Palubeckis instances of sizes 3000–7000. For practical use of the solver, we show the dependence of the time to best solution on the desired gap to the best known solution. In addition, we study the dependence of the gap and the time to best solution on the size of the problems solved by the underlying optimizer. Our results were obtained by simulation, using a tabu 1-opt solver, due to the huge number of runs required and limited quantum annealer time availability.


Quantum annealing Quadratic unconstrained binary optimization Combinatoric optimization k-opt Local search Iterative heuristic solver 



The authors would like to thank Marko Bucyk for editing a draft of this paper and Robyn Foerster, Phil Goddard, and Pooya Ronagh for their useful comments. This work was supported by 1QB Information Technologies (1QBit) and Mitacs.

Compliance with ethical standards

Conflicts of interest

EH declares no conflict of interest. GR, BW, and MV were academic interns at 1QBit when the work was done, and GR and BW are still at 1QBit. 1QBit is focused on solving real-world problems using quantum computers. D-Wave Systems is a minority investor in 1QBit.


  1. 1.
    Boros, E., Prékopa, A.: Probabilistic bounds and algorithms for the maximum satisfiability problem. Ann. Oper. Res. 21(1–4), 109–126 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boros, E., Hammer, P.L.: Pseudo-Boolean optimization. Discrete Appl. Math. 123(1–3) pp. 155–225 (2002). Workshop on Discrete Optimization, DO’99, PiscatawayGoogle Scholar
  3. 3.
    Bourjolly, J.-M.: A quadratic 0-1 optimization algorithm for the maximum clique and stable set problems. Technical Report University of Michigan, Ann Arbor (1994)Google Scholar
  4. 4.
    Du, D.-Z., Pardalos, P.M.: Handbook of Combinatorial Optimization: Supplement, vol. 1. Springer, New York (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Pardalos, P.M., Rodgers, G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45(2), 131–144 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pardalos, P.M., Rodgers, G.P.: A branch and bound algorithm for the maximum clique problem. Comput. Oper. Res. 19(5), 363–375 (1992)CrossRefzbMATHGoogle Scholar
  7. 7.
    Pardalos, P.M., Xue, J.: The maximum clique problem. J. Global Optim. 4(3), 301–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kochenberger, G.A., Glover, F., Alidaee, B., Rego, C.: A unified modeling and solution framework for combinatorial optimization problems. OR Spect. 26(2), 237–250 (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: An application of combinatorial optimization to statistical physics and circuit layout design. Oper. Res. 36(3), 493–513 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    De Simone, C., Diehl, M., Jünger, M., Mutzel, P., Reinelt, G., Rinaldi, G.: Exact ground states of ising spin glasses: New experimental results with a branch-and-cut algorithm. J. Stat. Phys. 80(1–2), 487–496 (1995)CrossRefzbMATHGoogle Scholar
  11. 11.
    Alidaee, B., Kochenberger, G.A., Ahmadian, A.: 0–1 quadratic programming approach for optimum solutions of two scheduling problems. Int. J. Syst. Sci. 25(2), 401–408 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. Handbook of Combinatorial Optimization, pp. 1–74. Springer, New York (1999)CrossRefGoogle Scholar
  13. 13.
    Iasemidis, L.D., Pardalos, P., Sackellares, J.C., Shiau, D.-S.: Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Comb. Optim. 5(1), 9–26 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Alidaee, B., Glover, F., Kochenberger, G.A., Rego, C.: A new modeling and solution approach for the number partitioning problem. J. Appl. Math. Decis. Sci. 2005(2), 113–121 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gulati, V., Gupta, S., Mittal, A.: Unconstrained quadratic bivalent programming problem. Eur. J. Oper. Res. 15(1), 121–125 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Carter, M.W.: The indefinite zero-one quadratic problem. Discr. Appl. Math. 7(1), 23–44 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Williams, H.P.: Model Building in Linear and Integer Programming. Springer, Berlin (1985)CrossRefzbMATHGoogle Scholar
  18. 18.
    Barahona, F., Jünger, M., Reinelt, G.: Experiments in quadratic 0–1 programming. Math. Prog. 44(1–3), 127–137 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Pardalos, P.M., Rodgers, G.P.: Parallel branch and bound algorithms for quadratic zero–one programs on the hypercube architecture. Ann. Oper. Res. 22(1), 271–292 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Billionnet, A., Sutter, A.: Minimization of a quadratic pseudo-boolean function. Eur. J. Oper. Res. 78(1), 106–115 (1994)CrossRefzbMATHGoogle Scholar
  21. 21.
    Palubeckis, G.: A heuristic-based branch and bound algorithm for unconstrained quadratic zero-one programming. Computing 54(4), 283–301 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Helmberg, C., Rendl, F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Prog. 82(3), 291–315 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hansen, P., Jaumard, B., Meyer, C., Groupe, Q.: d’études et de recherche en analyse des décisions (Montréal, Exact sequential algorithms for additive clustering. Montréal: Groupe d’études et de recherche en analyse des décisions (2000)Google Scholar
  24. 24.
    Huang, H.-X., Pardalos, P., Prokopyev, O.: Lower bound improvement and forcing rule for quadratic binary programming. Comput. Optim. Appl. 33(2–3), 187–208 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pardalos, P.M., Prokopyev, O.A., Busygin, S.: Continuous approaches for solving discrete optimization problems. In: Handbook on Modelling for Discrete Optimization, pp. 39–60, Springer (2006)Google Scholar
  26. 26.
    Pan, S., Tan, T., Jiang, Y.: A global continuation algorithm for solving binary quadratic programming problems. Comput. Optim. Appl. 41(3), 349–362 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gueye, S., Michelon, P.: A linearization framework for unconstrained quadratic (0–1) problems. Discret. Appl. Math. 157(6), 1255–1266 (2009). Reformulation Techniques and Mathematical ProgrammingMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pham Dinh, T., Nguyen Canh, N., Le Thi, H.: An efficient combined dca and bnb using dc/sdp relaxation for globally solving binary quadratic programs. J. Global Optim. 48(4), 595–632 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mauri, G.R., Lorena, L.A.N.: Lagrangean decompositions for the unconstrained binary quadratic programming problem. Int. Trans. Oper. Res. 18(2), 257–270 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mauri, G.R., Lorena, L.A.N.: A column generation approach for the unconstrained binary quadratic programming problem. Eur. J. Oper. Res. 217(1), 69–74 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mauri, G.R., Lorena, L.A.N.: Improving a lagrangian decomposition for the unconstrained binary quadratic programming problem. Comput. Oper. Res. 39(7), 1577–1581 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, D., Sun, X., Liu, C.: An exact solution method for unconstrained quadratic 0–1 programming: a geometric approach. J. Glob. Optim. 52(4), 797–829 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Beasley, J.: Heuristic algorithms for the unconstrained binary quadratic programming problem (1998)Google Scholar
  34. 34.
    Glover, F., Kochenberger, G.A., Alidaee, B.: Adaptive memory tabu search for binary quadratic programs. Manag. Sci. 44(3), 336–345 (1998)CrossRefzbMATHGoogle Scholar
  35. 35.
    Glover, F., Kochenberger, G., Alidaee, B., Amini, M.: Meta-Heuristics. Tabu search with critical event memory: an enhanced application for binary quadratic programs, pp. 93–109. Springer, New York (1999)Google Scholar
  36. 36.
    Palubeckis, G.: Iterated tabu search for the unconstrained binary quadratic optimization problem. Informatica 17(2), 279–296 (2006)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Glover, F., Lü, Z., Hao, J.-K.: Diversification-driven tabu search for unconstrained binary quadratic problems. 4OR 8(3), 239–253 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Lü, Z., Hao, J.-K., Glover, F.: Evolutionary Computation in Combinatorial Optimization. A study of memetic search with multi-parent combination for UBQP, pp. 154–165. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  39. 39.
    Shylo, V., Shylo, O.: Systems analysis; solving unconstrained binary quadratic programming problem by global equilibrium search. Cyber. Syst. Anal. 47(6), 889–897 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Lü, Z., Hao, J.-K., Glover, F.: Neighborhood analysis: a case study on curriculum-based course timetabling. J. Heuristics 17(2), 97–118 (2011)CrossRefGoogle Scholar
  41. 41.
    Wang, Y., Lü, Z., Glover, F., Hao, J.-K.: Probabilistic grasp-tabu search algorithms for the UBQP problem. Comput. Oper. Res. 40(12), 3100–3107 (2013)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Alkhamis, T.M., Hasan, M., Ahmed, M.A.: Simulated annealing for the unconstrained quadratic pseudo-boolean function. Eur. J. Oper. Res. 108(3), 641–652 (1998)CrossRefzbMATHGoogle Scholar
  43. 43.
    Katayama, K., Narihisa, H.: Performance of simulated annealing-based heuristic for the unconstrained binary quadratic programming problem. Eur. J. Oper. Res. 134(1), 103–119 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Merz, P., Freisleben, B.: Genetic algorithms for binary quadratic programming. In: Proceedings of the Genetic and Evolutionary Computation Conference, vol. 1, pp. 417–424, Citeseer, (1999)Google Scholar
  45. 45.
    Katayama, K., Tani, M., Narihisa, H.: Solving large binary quadratic programming problems by effective genetic local search algorithm. In: GECCO, pp. 643–650 (2000)Google Scholar
  46. 46.
    Lodi, A., Allemand, K., Liebling, T.M.: An evolutionary heuristic for quadratic 0–1 programming. Eur. J. Oper. Res. 119(3), 662–670 (1999)CrossRefzbMATHGoogle Scholar
  47. 47.
    Cai, Y., Wang, J., Yin, J., Zhou, Y.: Memetic clonal selection algorithm with \(\{{\rm EDA}\}\) vaccination for unconstrained binary quadratic programming problems. Expert Syst. Appl. 38(6), 7817–7827 (2011)CrossRefGoogle Scholar
  48. 48.
    Wang, Y., Lü, Z., Glover, F., Hao, J.-K.: A multilevel algorithm for large unconstrained binary quadratic optimization. In: Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems, pp. 395–408, Springer (2012)Google Scholar
  49. 49.
    Amini, M.M., Alidaee, B., Kochenberger, G.A.: New Ideas in Optimization. A scatter search approach to unconstrained quadratic binary programs, pp. 317–330. McGraw-Hill, New York (1999)Google Scholar
  50. 50.
    Palubeckis, G., Tomkevicius, A.: \(\{{\rm GRASP}\}\) implementations for the unconstrained binary quadratic optimization problem. Inf. Technol. Control 24, 14–20 (2002)Google Scholar
  51. 51.
    Boros, E., Hammer, P.L., Tavares, G.: Local search heuristics for quadratic unconstrained binary optimization (QUBO). J. Heuristics 13(2), 99–132 (2007)CrossRefGoogle Scholar
  52. 52.
    Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization. Technical Report Rutcor (2006)Google Scholar
  53. 53.
    Dantzig, G.B., Wolfe, P.: Decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)CrossRefzbMATHGoogle Scholar
  54. 54.
    Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)zbMATHGoogle Scholar
  55. 55.
    Boros, E., Hammer, P.: A max-flow approach to improved roof duality in quadratic 0-1 minimization. Rutgers University. Rutgers Center for Operations Research (RUTCOR) (1989)Google Scholar
  56. 56.
    Chardaire, P., Sutter, A.: A decomposition method for quadratic zero-one programming. Manag. Sci. 41(4), 704–712 (1995)CrossRefzbMATHGoogle Scholar
  57. 57.
    Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization (2006)Google Scholar
  58. 58.
    Bian, Z., Chudak, F., Israel, R., Lackey, B., Macready, W.G., Roy, A.: Discrete optimization using quantum annealing on sparse Ising models. Interdiscip. Phys. 2, 56 (2014)Google Scholar
  59. 59.
    Zintchenko, I., Hastings, M.B., Troyer, M.: From local to global ground states in ising spin glasses. Phys. Rev. B 91(2), 24201 (2015)CrossRefGoogle Scholar
  60. 60.
    Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58(5), 5355 (1998)CrossRefGoogle Scholar
  61. 61.
    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(5), 343–348 (1994)CrossRefGoogle Scholar
  62. 62.
    Ray, P., Chakrabarti, B.K., Chakrabarti, A.: Sherrington-kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations. Phys. Rev. B 39(16), 11828 (1989)CrossRefGoogle Scholar
  63. 63.
    Santoro, G.E., Martoňák, R., Tosatti, E., Car, R.: Theory of quantum annealing of an Ising spin glass. Science 295(5564), 2427–2430 (2002)CrossRefGoogle Scholar
  64. 64.
    Martoňák, R., Santoro, G.E., Tosatti, E.: Quantum annealing by the path-integral monte carlo method: the two-dimensional random ising model. Phys. Rev. B 66, 094203 (2002)CrossRefGoogle Scholar
  65. 65.
    Crosson, E., Harrow, A. W.: Simulated quantum annealing can be exponentially faster than classical simulated annealing, arXiv preprint arXiv:1601.03030 (2016)
  66. 66.
    Battaglia, D.A., Santoro, G.E., Tosatti, E.: Optimization by quantum annealing: lessons from hard satisfiability problems. Phys. Rev. E 71(6), 066707 (2005)CrossRefGoogle Scholar
  67. 67.
    Lanting, T., Przybysz, A.J., Smirnov, A.Y., Spedalieri, F.M., Amin, M.H., Berkley, A.J., Harris, R., Altomare, F., Boixo, S., Bunyk, P., et al.: Entanglement in a quantum annealing processor. Phys. Rev. X 4(2), 021041 (2014)Google Scholar
  68. 68.
    Neven, H., Smelyanskiy, V. N., Boixo, S., Shabani, A., Isakov, S. V., Dykman, M., Denchev, V. S., Amin, M., Smirnov, A., Mohseni, M.: Computational role of collective tunneling in a quantum annealer. Bull. Am. Phys. Soc. 60(1) (2015)Google Scholar
  69. 69.
    McGeoch, C. C., Wang, C.: Experimental evaluation of an adiabiatic quantum system for combinatorial optimization. In: Proceedings of the ACM International Conference on Computing Frontiers, ACM (2013)Google Scholar
  70. 70.
    Katzgraber, H.G., Hamze, F., Andrist, R.S.: Glassy chimeras could be blind to quantum speedup: designing better benchmarks for quantum annealing machines. Phys. Rev. X 4(2), 021008 (2014)Google Scholar
  71. 71.
    Boixo, S., Rønnow, T.F., Isakov, S.V., Wang, Z., Wecker, D., Lidar, D.A., Martinis, J.M., Troyer, M.: Evidence for quantum annealing with more than one hundred qubits. Nat. Phys. 10(3), 218–224 (2014)CrossRefGoogle Scholar
  72. 72.
    Hen, I., Job, J., Albash, T., Rønnow, T.F., Troyer, M., Lidar, D.A.: Probing for quantum speedup in spin-glass problems with planted solutions. Phys. Rev. A 92(4), 042325 (2015)CrossRefGoogle Scholar
  73. 73.
    King, A.D.: Performance of a quantum annealer on range-limited constraint satisfaction problems, arXiv preprint. arXiv:1502.02098 (2015)
  74. 74.
    Denchev, V.S., Boixo, S., Isakov, S.V., Ding, N., Babbush, R., Smelyanskiy, V., Martinis, J., Neven, H.: What is the computational value of finite range tunneling?, arXiv preprint. arXiv:1512.02206 (2015)
  75. 75.
    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(6195), 420–424 (2014)CrossRefGoogle Scholar
  76. 76.
    Martin-Mayor, V., Hen, I.: Unraveling quantum annealers using classical hardness. Sci. Rep. 5 (2015)Google Scholar
  77. 77.
    Katzgraber, H.G., Hamze, F., Zhu, Z., Ochoa, A.J., Munoz-Bauza, H.: Seeking quantum speedup through spin glasses: the good, the bad, and the ugly. Phys. Rev. X 5(3), 031026 (2015)Google Scholar
  78. 78.
    Bunyk, P., Hoskinson, E.M., Johnson, M., Tolkacheva, E., Altomare, F., Berkley, A.J., Harris, R., Hilton, J.P., Lanting, T., Przybysz, A.J., Whittaker, J.: Architectural considerations in the design of a superconducting quantum annealing processor. IEEE Trans. Appl. Supercond. 24, 1–10 (2014)CrossRefGoogle Scholar
  79. 79.
    Choi, V.: Minor-embedding in adiabatic quantum computation: I. the parameter setting problem. Quant. Inf. Process. 7(5), 193–209 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quant. Inf. Process. 10(3), 343–353 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Williams, C.: State-of-the-art quantum annealing and its application to cryptology, Isaac Newton Institute (2014).
  82. 82.
    Merz, P., Katayama, K.: Memetic algorithms for the unconstrained binary quadratic programming problem. BioSystems 78(1), 99–118 (2004)CrossRefGoogle Scholar
  83. 83.
    Glover, F., Hao, J.-K.: Efficient evaluations for solving large 0–1 unconstrained quadratic optimisation problems. Int. J. Metaheuristics 1(1), 3–10 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Wang, Y., Lü, Z., Glover, F., Hao, J.-K.: Path relinking for unconstrained binary quadratic programming. Eur. J. Oper. Res. 223(3), 595–604 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  85. 85.
    Beasley, J.E.: OR-Library: Unconstrained binary quadratic programming,” (2014).
  86. 86.
    Tavares, G.: New algorithms for Quadratic Unconstrained Binary Optimization (QUBO) with applications in engineering and social sciences. PhD thesis, Rutgers University, Graduate School - New Brunswick (2008)Google Scholar
  87. 87.
    Palubeckis, G.: Multistart tabu search strategies for the unconstrained binary quadratic optimization problem. Ann. Oper. Res. 131(1–4), 259–282 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Pudenz, K.L., Albash, T., Lidar, D.A.: Error-corrected quantum annealing with hundreds of qubits. Nat. Commun. 5 (2014)Google Scholar
  89. 89.
    Pudenz, K.L., Albash, T., Lidar, D.A.: Quantum annealing correction for random ising problems. Phys. Rev. A 91, 042302 (2015)CrossRefGoogle Scholar
  90. 90.
    Barends, R., Lamata, L., Kelly, J., Garcia-Alvarez, L., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y., Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Hoi, I.C., Neill, C., O/’Malley, P.J.J., Quintana, C., Roushan, P., Vainsencher, A., Wenner, J., Solano, E., Martinis, J.M.: Digital quantum simulation of fermionic models with a superconducting circuit. Nat. Commun. 6, 07 (2015)Google Scholar
  91. 91.
    Lechner, W., Hauke, P., Zoller, P.: A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv. 1(9), e1500838 (2015)CrossRefGoogle Scholar
  92. 92.
    Pastawski, F., Preskill, J.: Error correction for a proposed quantum annealing architecture, arXiv preprint. arXiv:1511.00004 (2015)

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Gili Rosenberg
    • 1
    Email author
  • Mohammad Vazifeh
    • 1
  • Brad Woods
    • 1
  • Eldad Haber
    • 2
  1. 1.1QB Information Technologies (1QBit)VancouverCanada
  2. 2.Department of Mathematics and Earth and Ocean ScienceUniversity of British ColumbiaVancouverCanada

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