Boosting quantum annealer performance via sample persistence

Abstract

We propose a novel method for reducing the number of variables in quadratic unconstrained binary optimization problems, using a quantum annealer (or any sampler) to fix the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are usually much easier for the quantum annealer to solve, due to their being smaller and consisting of disconnected components. This approach significantly increases the success rate and number of observations of the best known energy value in samples obtained from the quantum annealer, when compared with calling the quantum annealer without using it, even when using fewer annealing cycles. Use of the method results in a considerable improvement in success metrics even for problems with high-precision couplers and biases, which are more challenging for the quantum annealer to solve. The results are further enhanced by applying the method iteratively and combining it with classical pre-processing. We present results for both Chimera graph-structured problems and embedded problems from a real-world application.

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

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    The chip at our disposal had 1100 active qubits, a working temperature of 26 ± 5 mK, and a minimum annealing time of 20 \(\upmu \)s.

  2. 2.

    A gauge, in this context, implies multiplying each spin operator by \(\pm 1\).

  3. 3.

    We used the function fix_variables in D-Wave Systems’ SAPI 2.3.1, which is the solver API used to access the quantum annealer [9].

  4. 4.

    The value zero was excluded for the couplers but not for the biases, and we use this convention throughout the paper.

  5. 5.

    The reduced problems often consist of multiple connected components (see Sect. 3.4). We took advantage of this fact when evaluating the energy values for the states in each sample.

References

  1. 1.

    Battaglia, D.A., Santoro, G.E., Tosatti, E.: Optimization by quantum annealing: lessons from hard satisfiability problems. Phys. Rev. E 71(6), 066,707 (2005)

    Article  Google Scholar 

  2. 2.

    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)

    Article  Google Scholar 

  3. 3.

    Boixo, S., Smelyanskiy, V.N., Shabani, A., Isakov, S.V., Dykman, M., Denchev, V.S., Amin, M.H., Smirnov, A.Y., Mohseni, M., Neven, H.: Computational multiqubit tunnelling in programmable quantum annealers. Nat. Commun. 7, 10327 (2016). doi:10.1038/ncomms10327

  4. 4.

    Boros, E., Hammer, P.L., Tavares, G.: Preprocessing of unconstrained quadratic binary optimization. Rutcor research report (2006)

  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., et al.: Architectural considerations in the design of a superconducting quantum annealing processor. IEEE Trans. Appl. Supercond. 24(4), 1–10 (2014)

    Article  Google Scholar 

  6. 6.

    Chardaire, P., Lutton, J.L., Sutter, A.: Thermostatistical persistency: a powerful improving concept for simulated annealing algorithms. Eur. J. Oper. Res. 86(3), 565–579 (1995)

    Article  MATH  Google Scholar 

  7. 7.

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

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    D-Wave Systems: Private communication (2016)

  10. 10.

    D-Wave Systems: SAPI 2.3.1 documentation (2016)

  11. 11.

    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? Phys. Rev. X 6, 031,015 (2016). doi:10.1103/PhysRevX.6.031015

    Google Scholar 

  12. 12.

    Finnila, A., Gomez, M., Sebenik, C., Stenson, C., Doll, J.: Quantum annealing: a new method for minimizing multidimensional functions. Chem. Phys. Lett. 219(5), 343–348 (1994)

    ADS  Article  Google Scholar 

  13. 13.

    Glover, F.: Tabu search-part I. ORSA J. Comput. 1(3), 190–206 (1989)

    Article  MATH  Google Scholar 

  14. 14.

    Glover, F.: Tabu search-part II. ORSA J. Comput. 2(1), 4–32 (1990)

    Article  MATH  Google Scholar 

  15. 15.

    Hammer, P.L., Hansen, P., Simeone, B.: Roof duality, complementation and persistency in quadratic 0–1 optimization. Math. Program. 28(2), 121–155 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Hamze, F., de Freitas, N.: From fields to trees. In: Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, pp. 243–250. AUAI Press (2004)

  17. 17.

    Heim, B., Rønnow, T.F., Isakov, S.V., Troyer, M.: Quantum versus classical annealing of ising spin glasses. Science 348(6231), 215–217 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    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), 042,325 (2015)

    Article  Google Scholar 

  19. 19.

    Jiang, H., Xuan, J.: Backbone guided local search for the weighted maximum satisfiability problem. INTECH Open Access Publisher (2009)

  20. 20.

    Johnson, M., Amin, M., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Harris, R., Berkley, A., Johansson, J., Bunyk, P., et al.: Quantum annealing with manufactured spins. Nature 473(7346), 194–198 (2011)

    ADS  Article  Google Scholar 

  21. 21.

    Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58(5), 5355 (1998)

    ADS  Article  Google Scholar 

  22. 22.

    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), 021,008 (2014)

    Google Scholar 

  23. 23.

    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), 031,026 (2015)

    Google Scholar 

  24. 24.

    King, A.D.: Performance of a quantum annealer on range-limited constraint satisfaction problems. arXiv preprint arXiv:1502.02098 (2015)

  25. 25.

    King, J., Yarkoni, S., Nevisi, M.M., Hilton, J.P., McGeoch, C.C.: Benchmarking a quantum annealing processor with the time-to-target metric. arXiv preprint arXiv:1508.05087 (2015)

  26. 26.

    Lanting, T., Przybysz, A., Smirnov, A.Y., Spedalieri, F., Amin, M., Berkley, A., 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 

  27. 27.

    Lechner, W., Hauke, P., Zoller, P.: A quantum annealing architecture with all-to-all connectivity from local interactions. Sci. Adv. 1(9), e1500,0838 (2015)

    Article  Google Scholar 

  28. 28.

    Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2(5) (2014). doi:10.3389/fphy.2014.00005. http://www.frontiersin.org/interdisciplinary_physics/10.3389/fphy.2014.00005/abstract

  29. 29.

    Mandrà, S., Zhu, Z., Wang, W., Perdomo-Ortiz, A., Katzgraber, H.G.: Strengths and weaknesses of weak-strong cluster problems: a detailed overview of state-of-the-art classical heuristics versus quantum approaches. Phys. Rev. A 94, 022,337 (2016). doi:10.1103/PhysRevA.94.022337

    Article  Google Scholar 

  30. 30.

    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, p. 23. ACM (2013)

  31. 31.

    Mishra, A., Albash, T., Lidar, D.A.: Performance of two different quantum annealing correction codes. Quantum Inf. Process. 15(2), 609–636 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic ‘phase transitions’. Nature 400(6740), 133–137 (1999)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Muthukrishnan, S., Albash, T., Lidar, D.A.: Tunneling and speedup in quantum optimization for permutation-symmetric problems. Phys. Rev. X 6, 031,010 (2016). doi:10.1103/PhysRevX.6.031010

    Google Scholar 

  34. 34.

    Pastawski, F., Preskill, J.: Error correction for encoded quantum annealing. Phys. Rev. A 93, 052,325 (2016). doi:10.1103/PhysRevA.93.052325

    Article  Google Scholar 

  35. 35.

    Perdomo-Ortiz, A., Fluegemann, J., Biswas, R., Smelyanskiy, V.N.: A performance estimator for quantum annealers: Gauge selection and parameter setting. arXiv preprint arXiv:1503.01083 (2015)

  36. 36.

    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, 18,628 (2016)

    Article  Google Scholar 

  37. 37.

    Pudenz, K.L., Albash, T., Lidar, D.A.: Error-corrected quantum annealing with hundreds of qubits. Nat. Commun. 5, 3243 (2014). doi:10.1038/ncomms4243

  38. 38.

    Pudenz, K.L., Albash, T., Lidar, D.A.: Quantum annealing correction for random Ising problems. Phys. Rev. A 91(4), 042,302 (2015)

    Article  Google Scholar 

  39. 39.

    Ray, P., Chakrabarti, B., Chakrabarti, A.: Sherrington-Kirkpatrick model in a transverse field: absence of replica symmetry breaking due to quantum fluctuations. Phys. Rev. B 39(16), 11,828 (1989)

    Article  Google Scholar 

  40. 40.

    Romá, F., Risau-Gusman, S.: Backbone structure of the Edwards–Anderson spin-glass model. Phys. Rev. E 88(4), 042,105 (2013)

    Article  Google Scholar 

  41. 41.

    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). doi:10.1126/science.1252319. http://www.sciencemag.org/content/345/6195/420.abstract

  42. 42.

    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)

  43. 43.

    Rosenberg, G., Vazifeh, M., Woods, B., Haber, E.: Building an iterative heuristic solver for a quantum annealer. Comput. Optim. Appl. 65(13), 845–869 (2015)

  44. 44.

    Santoro, G.E., Martoňák, R., Tosatti, E., Car, R.: Theory of quantum annealing of an Ising spin glass. Science 295(5564), 2427–2430 (2002)

    ADS  Article  Google Scholar 

  45. 45.

    Selby, A.: QUBO-Chimera. https://github.com/alex1770/QUBO-Chimera (2013)

  46. 46.

    Selby, A.: Efficient subgraph-based sampling of Ising-type models with frustration. arXiv preprint arXiv:1409.3934 (2014)

  47. 47.

    Tavares, G.: New algorithms for Quadratic Unconstrained Binary Optimization (QUBO) with applications in engineering and social sciences. Ph.D. thesis, Rutgers University, Graduate School—New Brunswick (2008). doi:10.7282/T3XK8FS2

  48. 48.

    Tran, T.T., Do, M., Rieffel, E.G., Frank, J., Wang, Z., O’Gorman, B., Venturelli, D., Beck, J.C.: A hybrid quantum-classical approach to solving scheduling problems. In: Ninth Annual Symposium on Combinatorial Search (2016)

  49. 49.

    Vinci, W., Albash, T., Paz-Silva, G., Hen, I., Lidar, D.A.: Quantum annealing correction with minor embedding. Phys. Rev. A 92(4), 042310 (2015)

    ADS  Article  Google Scholar 

  50. 50.

    Wang, Y., Lü, Z., Glover, F., Hao, J.K.: Effective variable fixing and scoring strategies for binary quadratic programming. In: Evolutionary Computation in Combinatorial Optimization, pp. 72–83. Springer (2011)

  51. 51.

    Wang, Y., Lü, Z., Glover, F., Hao, J.K.: Backbone guided tabu search for solving the UBQP problem. J. Heuristics 19(4), 679–695 (2013)

  52. 52.

    Zaribafiyan, A., Marchand, D.J.J., Changiz Rezaei, S.S.: Systematic and deterministic graph-minor embedding for Cartesian products of graphs. Quantum Inf. Process. 16(5), 136 (2017). doi:10.1007/s11128-017-1569-z

  53. 53.

    Zhang, W.: Configuration landscape analysis and backbone guided local search. Part I: satisfiability and maximum satisfiability. Artif. Intell. 158(1), 1–26 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  54. 54.

    Zintchenko, I., Hastings, M.B., Troyer, M.: From local to global ground states in Ising spin glasses. Phys. Rev. B 91(2), 024,201 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Dominic Marchand, Pooya Ronagh, and Brad Woods for their insightful comments, Marko Bucyk for editing the manuscript, and Alex Selby for the use of his implementation of the Hamze–de Freitas–Selby (HFS) algorithm, available for public use on GitHub [45]. This work was supported by 1QBit and Mitacs.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hamed Karimi.

Ethics declarations

Conflict of interest

Hamed Karimi is an academic intern, and Gili Rosenberg is an employee at 1QBit. 1QBit is focused on solving real-world problems using quantum computers. D-Wave Systems is a minority investor in 1QBit.

Appendices

Appendix 1: Dependence on thresholds

Table 6 Mean number of fixed variables for different fixing_threshold (column: ‘F’) and elite_threshold (row: ‘E’) using SPVAR on Chimera graph-structured problems (with 1100 variables)

To illustrate the dependence of the number of fixed variables and SPVAR’s fixing success rate on fixing_threshold and elite_threshold, we present results for two problem sets in Tables 6 and 7. The fixing success rate was defined as the percentage of problems for which the method fixed variables only to their optimal value. Detailed results for two choices of parameters, for the same two problem sets, are presented in Fig. 4. In that experiment, SPVAR was applied on a quantum annealer sample obtained from 2500 annealing cycles. We defined the quantum annealer’s residual as the energy difference between the quantum annealer’s lowest energy solution and the best known solution (found by a heuristic solver; see Sect. 3 for more details), and the method’s residual as the difference in energy between the best known solution before fixing variables and after fixing variables.

We see that the quantum annealer was able to find the best known solution for most of the low-precision problems (top row), but unable to find the best known solution for almost all of the higher-precision problems (bottom row). The method’s residual was almost always zero, indicating that the method almost never fixed variables incorrectly. In the few cases in which it did fix variables incorrectly, the effective problem (after fixing variables) still conserved the first or second excited states (the energy spacing was exactly two, due to the integer biases and couplers). We also note that for those problems, the method’s residual was always lower than the quantum annealer’s residual. The mean fraction of variables fixed was 58–70%, showing that SPVAR was able to fix most of the variables in these problems, and, as described in Sect. 3.2, the iterative method presented (ISPVAR) was able to fix even more variables.

Table 7 SPVAR fixing success rate for different fixing_threshold (column: ‘F’) and elite_threshold (row: ‘E’) on Chimera graph-structured problems
Fig. 4
figure4

(Colour online) Results for SPVAR on Chimera graph-structured problems. a and b Present results for problems in \(U_2\) (\(J,h \in \{-2,-1,0,1,2\}\), excluding 0 for J), and c and d present results for problems in \(U_{10}\). The parameters are indicated in the titles above each subfigure. The top panel of each subfigure shows the difference in energy between the quantum annealer’s best solution and the best known solution, for each problem, obtained from 2500 annealing cycles using five random gauges. The middle panel shows the difference in energy between the best known solution before fixing variables and the best known solution after fixing variables (using the same 2500 cycles)—it should be zero if all variables fixed by SPVAR were fixed correctly. The bottom panel shows the number of fixed variables for each problem. A horizontal dashed line indicates the mean number of fixed variables. In all panels, the points that correspond to problems for which SPVAR fixed variables incorrectly are shown in red, and the rest are shown in blue

Appendix 2: Parameter description

In Table 8, we list the parameters used in the text and give a short description of each.

Table 8 Parameters of SPVAR and ISPVAR

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Karimi, H., Rosenberg, G. Boosting quantum annealer performance via sample persistence. Quantum Inf Process 16, 166 (2017). https://doi.org/10.1007/s11128-017-1615-x

Download citation

Keywords

  • Adiabatic quantum computation
  • Quantum annealing
  • Variable reduction
  • Sample persistency
  • Binary optimization