Efficiently embedding QUBO problems on adiabatic quantum computers


Adiabatic quantum computers like the D-Wave 2000Q can approximately solve the QUBO problem, which is an NP-hard problem, and have been shown to outperform classical computers on several instances. Solving the QUBO problem literally means solving virtually any NP-hard problem like the traveling salesman problem, airline scheduling problem, protein folding problem, genotype imputation problem, thereby enabling significant scientific progress, and potentially saving millions/billions of dollars in logistics, airlines, healthcare and many other industries. However, before QUBO problems are solved on quantum computers, they must be embedded (or compiled) onto the hardware of quantum computers, which in itself is a very hard problem. In this work, we propose an efficient embedding algorithm, that lets us embed QUBO problems fast, uses less qubits and gets the objective function value close to the global minimum value. We then compare the performance of our embedding algorithm to that of D-Wave’s embedding algorithm, which is the current state of the art, and show that our embedding algorithm convincingly outperforms D-Wave’s embedding algorithm. Our embedding approach works with perfect Chimera graphs, i.e., Chimera graphs with no missing qubits.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    Adachi, S.H., Davenport, D.M., Henderson, M.P.: Quantum-assisted training of neural networks. US Patent App. 14/702,203 (2015)

  2. 2.

    Albash, T., Lidar, D.A.: Adiabatic quantum computation. Rev. Modern Phys. 90(1), 015002 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Amin, M.H.: Methods of adiabatic quantum computation. US Patent 8,504,497 (2013)

  4. 4.

    Amin, M.H., Steininger, M.F.: Adiabatic quantum computation with superconducting qubits. US Patent 7,135,701 (2006)

  5. 5.

    Applegate, D.L., Bixby, R.E., Chvatal, V., Cook, W.J.: The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  6. 6.

    Barends, R., Shabani, A., Lamata, L., Kelly, J., Mezzacapo, A., Las Heras, U., Babbush, R., Fowler, A.G., Campbell, B., Chen, Y., et al.: Digitized adiabatic quantum computing with a superconducting circuit. Nature 534(7606), 222 (2016)

    ADS  Article  Google Scholar 

  7. 7.

    Baruah, S., Bertogna, M., Buttazzo, G.: Multiprocessor Scheduling for Real-time Systems. Springer, Berlin (2015)

    MATH  Book  Google Scholar 

  8. 8.

    Biamonte, J.D., Berkley, A.J., Amin, M.: Physical realizations of a universal adiabatic quantum computer. US Patent 8,234,103 (2012)

  9. 9.

    Bian, Z., Chudak, F., Israel, R.B., Lackey, B., Macready, W.G., Roy, A.: Mapping constrained optimization problems to quantum annealing with application to fault diagnosis. Front. ICT 3, 14 (2016)

    Article  Google Scholar 

  10. 10.

    Bian, Z., Chudak, F., Macready, W., Roy, A., Sebastiani, R., Varotti, S.: Solving sat and maxsat with a quantum annealer: foundations and a preliminary report. In: International Symposium on Frontiers of Combining Systems, pp. 153–171. Springer, Berlin (2017)

  11. 11.

    Blum, A., Rivest, R.L.: Training a 3-node neural network is NP-complete. In: Advances in Neural Information Processing Systems, pp. 494–501 (1989)

  12. 12.

    Boothby, T., King, A.D., Roy, A.: Fast clique minor generation in chimera qubit connectivity graphs. Quantum Inf. Process. 15(1), 495–508 (2016)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Britt, K.A., Humble, T.S.: High-performance computing with quantum processing units. ACM J. Emerg. Technol. Comput. Syst. (JETC) 13(3), 39 (2017)

    Google Scholar 

  14. 14.

    Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors. Preprint (2014). arXiv:1406.2741

  15. 15.

    Carrillo, H., Lipman, D.: The multiple sequence alignment problem in biology. SIAM J. Appl. Math. 48(5), 1073–1082 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Chickering, D.M., Geiger, D., Heckerman, D., et al.: Learning Bayesian networks is NP-hard. Technical Report, Citeseer (1994)

  17. 17.

    Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7(5), 193–209 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Choi, V.: Minor-embedding in adiabatic quantum computation: II. Minor-universal graph design. Quantum Inf. Process. 10(3), 343–353 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Cook, S.A.: The complexity of theorem-proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pp. 151–158. ACM, New York (1971)

  20. 20.

    Cristianini, N., Shawe-Taylor, J., et al.: An Introduction to Support Vector Machines and Other Kernel-based Learning Methods. Cambridge University Press, Cambridge (2000)

    MATH  Book  Google Scholar 

  21. 21.

    Dasgupta, S.: The hardness of \(k\)-means clustering. Department of Computer Science and Engineering, University of California, San Diego (2008)

  22. 22.

    Dill, K.A., MacCallum, J.L.: The protein-folding problem, 50 years on. Science 338(6110), 1042–1046 (2012)

    ADS  Article  Google Scholar 

  23. 23.

    Dubois, O., Dequen, G.: A backbone-search heuristic for efficient solving of hard 3-sat formulae. IJCAI 1, 248–253 (2001)

    Google Scholar 

  24. 24.

    Esteve, D., Vion, D., Devoret, M., Urbina, C., Joyez, P., Pothier, H., Orfila, P.F., Aassime, A., Cottet, A.: Superconducting quantum-bit device based on josephson junctions. US Patent 6,838,694 (2005)

  25. 25.

    Etschmaier, M.M., Mathaisel, D.F.: Airline scheduling: an overview. Transp. Sci. 19(2), 127–138 (1985)

    Article  Google Scholar 

  26. 26.

    Goodrich, T.D., Sullivan, B.D., Humble, T.S.: Optimizing adiabatic quantum program compilation using a graph-theoretic framework. Quantum Inf. Process. 17(5), 118 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pp. 212–219. ACM, New York (1996)

  28. 28.

    Hamilton, K.E., Humble, T.S.: Identifying the minor set cover of dense connected bipartite graphs via random matching edge sets. Quantum Inf. Process. 16(4), 94 (2017)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Huang, W., Yu, J.X.: Investigating TSP heuristics for location-based services. Data Sci. Eng. 2(1), 71–93 (2017)

    ADS  Article  Google Scholar 

  30. 30.

    Imamog, A., Awschalom, D.D., Burkard, G., DiVincenzo, D.P., Loss, D., Sherwin, M., Small, A., et al.: Quantum information processing using quantum dot spins and cavity QED. Phys. Rev. Lett. 83(20), 4204 (1999)

    ADS  Article  Google Scholar 

  31. 31.

    Kane, B.E.: A silicon-based nuclear spin quantum computer. Nature 393(6681), 133 (1998)

    ADS  Article  Google Scholar 

  32. 32.

    Keating, T., Goyal, K., Jau, Y.Y., Biedermann, G.W., Landahl, A.J., Deutsch, I.H.: Adiabatic quantum computation with Rydberg-dressed atoms. Phys. Rev. A 87(5), 052314 (2013)

    ADS  Article  Google Scholar 

  33. 33.

    King, A.D., Israel, R.B., Bunyk, P.I., Boothby, T.J., Reinhardt, S.P., Roy, A.P., King, J.A., Lanting, T.M., Evert, A.J.: Systems and methods for embedding problems into an analog processor. US Patent App. 15/487,295 (2017)

  34. 34.

    Kleinberg, J., Tardos, E.: Algorithm Design. Pearson Education India, Noida (2006)

    Google Scholar 

  35. 35.

    Klymko, C., Sullivan, B.D., Humble, T.S.: Adiabatic quantum programming: minor embedding with hard faults. Quantum Inf. Process. 13(3), 709–729 (2014)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Landahl, A.: Adiabatic quantum computing. In: APS Four Corners Section Meeting Abstracts (2012)

  37. 37.

    Lewis, M., Glover, F.: Quadratic unconstrained binary optimization problem preprocessing: theory and empirical analysis. Networks 70(2), 79–97 (2017)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Li, Y., Willer, C., Sanna, S., Abecasis, G.: Genotype imputation. Ann. Rev. Genom. Hum. Genet. 10, 387–406 (2009)

    Article  Google Scholar 

  39. 39.

    Macready, W., Roy, A.P.: Systems and methods that formulate problems for solving by a quantum processor using hardware graph decomposition. US Patent 9,875,215 (2018)

  40. 40.

    Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. AAPT (2002)

  41. 41.

    Papadimitriou, C.H.: Computational Complexity. Wiley, London (2003)

    MATH  Google Scholar 

  42. 42.

    Pittman, T., Jacobs, B., Franson, J.: Probabilistic quantum logic operations using polarizing beam splitters. Phys. Rev. A 64(6), 062311 (2001)

    ADS  Article  Google Scholar 

  43. 43.

    Pla, J.J., Tan, K.Y., Dehollain, J.P., Lim, W.H., Morton, J.J., Jamieson, D.N., Dzurak, A.S., Morello, A.: A single-atom electron spin qubit in silicon. Nature 489(7417), 541 (2012)

    ADS  Article  Google Scholar 

  44. 44.

    Potok, T.E., Schuman, C.D., Young, S.R., Patton, R.M., Spedalieri, F., Liu, J., Yao, K.T., Rose, G., Chakma, G.: A study of complex deep learning networks on high performance, neuromorphic, and quantum computers. In: Workshop on Machine Learning in HPC Environments (MLHPC), pp. 47–55. IEEE, New York (2016)

  45. 45.

    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)

    ADS  MATH  Article  Google Scholar 

  46. 46.

    Roy, A.P.: Systems and methods that formulate embeddings of problems for solving by a quantum processor. US Patent 9,501,747 (2016)

  47. 47.

    Shmygelska, A., Hoos, H.H.: An ant colony optimisation algorithm for the 2D and 3D hydrophobic polar protein folding problem. BMC Bioinformat. 6(1), 30 (2005)

    Article  Google Scholar 

  48. 48.

    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: 1994 Proceedings., 35th Annual Symposium on Foundations of Computer Science, pp. 124–134. IEEE, New York (1994)

  49. 49.

    Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Rev. 41(2), 303–332 (1999)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Thom, M.C., Roy, A.P., Chudak, F.A., Bian, Z., Macready, W.G., Israel, R.B., Boothby, T.J., Yarkoni, S., Xue, Y., Korenkevych, D., et al.: Systems and methods for analog processing of problem graphs having arbitrary size and/or connectivity. US Patent App. 15/448,361 (2017)

  51. 51.

    Turing, A.M.: On computable numbers, with an application to the entscheidungsproblem. a correction. Proc. Lond. Math. Soc. 2(1), 544–546 (1938)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Venturelli, D., Mandra, S., Knysh, S., O’Gorman, B., Biswas, R., Smelyanskiy, V.: Quantum optimization of fully connected spin glasses. Phys. Rev. X 5(3), 031040 (2015)

    Google Scholar 

  53. 53.

    Zagoskin, A.M.: Qubit using a josephson junction between s-wave and d-wave superconductors. US Patent 6,459,097 (2002)

  54. 54.

    Zagoskin, A.M.: Quantum computing method using magnetic flux states at a Josephson junction. US Patent 6,563,311 (2003)

  55. 55.

    Zaribafiyan, A., Marchand, D., REZAEI, S.S.C.: Method and system for generating an embedding pattern used for solving a quadratic binary optimization problem. US Patent App. 15/344,054 (2017)

Download references

Author information



Corresponding author

Correspondence to Prasanna Date.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported in part by an appointment to the Oak Ridge National Laboratory ASTRO Program, sponsored by the US Department of Energy and administered by the Oak Ridge Institute for Science and Education.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Date, P., Patton, R., Schuman, C. et al. Efficiently embedding QUBO problems on adiabatic quantum computers. Quantum Inf Process 18, 117 (2019). https://doi.org/10.1007/s11128-019-2236-3

Download citation


  • Adiabatic quantum computing
  • Embedding
  • Quadratic unconstrained binary optimization (QUBO)