Practical integer-to-binary mapping for quantum annealers

  • Sahar KarimiEmail author
  • Pooya Ronagh


Recent advancements in quantum annealing hardware and numerous studies in this area suggest that quantum annealers have the potential to be effective in solving unconstrained binary quadratic programming problems. Naturally, one may desire to expand the application domain of these machines to problems with general discrete variables. In this paper, we explore the possibility of employing quantum annealers to solve unconstrained quadratic programming problems over a bounded integer domain. We present an approach for encoding integer variables into binary ones, thereby representing unconstrained integer quadratic programming problems as unconstrained binary quadratic programming problems. To respect some of the limitations of the currently developed quantum annealers, we propose an integer encoding, named bounded-coefficient encoding, in which we limit the size of the coefficients that appear in the encoding. Furthermore, we propose an algorithm for finding the upper bound on the coefficients of the encoding using the precision of the machine and the coefficients of the original integer problem. We experimentally show that this approach is far more resilient to the noise of the quantum annealers compared to traditional approaches for the encoding of integers in base two. In addition, we perform time-to-solution analysis of various integer encoding strategies with respect to the size of integer programming problems and observe favorable performance from the bounded-coefficient encoding relative to that of the unary and binary encodings.


Adiabatic quantum computation Simulated quantum annealing Integer programming Integer encoding Bounded-coefficient encoding 



We are thankful to Helmut G. Katzgraber, Gili Rosenberg, and the referees of Quantum Information Processing for insightful discussions and helpful feedback. We would also like to thank Marko Bucyk for editing this manuscript.

Supplementary material


  1. 1.
    Albash, T., Lidar, D.A.: Adiabatic quantum computing (2016). arXiv:quant-ph/1611.04471
  2. 2.
    Albash, T., Vinci, W., Mishra, A., Warburton, P.A., Lidar, D.A.: Consistency tests of classical and quantum models for a quantum annealer. Phys. Rev. A 91, 042314 (2015). ADSCrossRefGoogle Scholar
  3. 3.
    Amin, M., Dickson, N., Smith, P.: Adiabatic quantum optimization with qudits. Quantum Inf. Process. 12(4), 1819–1829 (2013). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Aramon, M., Rosenberg, G., Valiante, E., Miyazawa, T., Tamura, H., Katzgraber, H.G.: Physics-inspired optimization for quadratic unconstrained problems using a digital annealer (2018). arXiv:1806.08815
  5. 5.
    Bunyk, P.I., Hoskinson, E.M., Johnson, M.W., 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(4), 1–10 (2014). CrossRefGoogle Scholar
  6. 6.
    Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors (2014). arXiv:1406.2741
  7. 7.
    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). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gurobi Optimization, Inc.: Gurobi Optimizer Reference Manual (2015).
  9. 9.
    Ishikawa, H.: Transformation of general binary MRF minimization to the first-order case. IEEE Trans. Pattern Anal. Mach. Intell. 33(6), 1234–1249 (2011). CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Heim, B., Rønnow, T.F., Isakov, S.V., Troyer, M.: Quantum versus classical annealing of Ising spin glasses. Science 348, 215–217 (2015)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Karimi, S., Ronagh, P.: A subgradient approach for constrained binary programming via quantum adiabatic evolution (2017). arXiv:1605.09462
  13. 13.
    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, 031026 (2015). CrossRefGoogle Scholar
  14. 14.
    Lee, L.W., Katzgraber, H.G., Young, A.P.: Critical behavior of the three- and ten-state short-range Potts glass: a Monte Carlo study. Phys. Rev. B 74, 104416 (2006). ADSCrossRefGoogle Scholar
  15. 15.
    Mandrà, S., Zhu, Z., Katzgraber, H.G.: Exponentially biased ground-state sampling of quantum annealing machines with transverse-field driving Hamiltonians. Phys. Rev. Lett. 118, 070502 (2017). ADSCrossRefGoogle Scholar
  16. 16.
    Mansini, R., Ogryczak, W., Speranza, M.G.: Linear Models for Portfolio Optimization, pp. 19–45. Springer, Berlin (2015). CrossRefzbMATHGoogle Scholar
  17. 17.
    Matsuda, Y., Nishimori, H., Katzgraber, H.G.: Ground-state statistics from annealing algorithms: quantum versus classical approaches. New J. Phys. 11(7), 073021 (2009). ADSCrossRefGoogle Scholar
  18. 18.
    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, CF ’13, New York, NY, USA, pp. 23:1–23:11 (2013). ACM.
  19. 19.
    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, 1–36 (2015). ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Rosenberg, G., Haghnegahdar, P., Goddard, P., Carr, P., Wu, K., de Prado, M.L.: Solving the optimal trading trajectory problem using a quantum annealer. IEEE J. Sel. Top. Signal Process. 10(6), 1053–1060 (2016). ADSCrossRefGoogle Scholar
  21. 21.
    Santoro, G.E., Tosatti, E.: Optimization using quantum mechanics: quantum annealing through adiabatic evolution. J. Phys. A Math. Gen. 39(36), R393 (2006). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sawik, T.: Scheduling in Supply Chains Using Mixed Integer Programming. Wiley, Hoboken (2011). CrossRefzbMATHGoogle Scholar
  23. 23.
    Venturelli, D., Marchand, D.J.J., Rojo, G.: Quantum annealing implementation of job-shop scheduling (2015). arXiv:1506.08479
  24. 24.
    Zhu, Z., Ochoa, A.J., Schnabel, S., Hamze, F., Katzgraber, H.G.: Best-case performance of quantum annealers on native spin-glass benchmarks: how chaos can affect success probabilities. Phys. Rev. A 93, 012317 (2016). ADSCrossRefGoogle Scholar
  25. 25.
    Zick, K.M., Shehab, O., French, M.: Experimental quantum annealing: case study involving the graph isomorphism problem. Sci. Rep. 5, 11168 (2015). ADSCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.1QB Information Technologies (1QBit)VancouverCanada
  2. 2.Institute for Quantum Computing (IQC)University of WaterlooWaterlooCanada

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