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Practical Implementation of a Quantum Backtracking Algorithm

  • Simon Martiel
  • Maxime RemaudEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12011)

Abstract

In previous work, Montanaro presented a method to obtain quantum speedups for backtracking algorithms, a general meta-algorithm to solve constraint satisfaction problems (CSPs). In this work, we derive a space efficient implementation of this method. Assume that we want to solve a CSP with m constraints on n variables and that the domain in which these variables take their value is of cardinality d. Then, we show that the implementation of Montanaro’s backtracking algorithm can be done by using \(\mathcal {O}(n\log {d})\) data qubits. We detail an implementation of the predicate associated to the CSP with an additional register of \(\mathcal {O}(\log {m})\) qubits. We explicit our implementation for graph coloring and SAT problems, and present simulation results. Finally, we discuss the impact of the usage of static and dynamic variable ordering heuristics in the quantum setting.

Keywords

Backtracking algorithm Quantum walk CSP Graph coloring SAT 

Notes

Acknowledgments

This work was supported by Atos. The implementation was developed in python using Atos’ pyAQASM library. All simulations were performed on the Atos Quantum Learning Machine. We acknowledge support from the French ANR project ANR-18-CE47-0010 (QUDATA), the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union’s Horizon 2020 Program (QuantAlgo project), and the French ANR project ANR-18-QUAN-0017 (QuantAlgo Project).

References

  1. 1.
    Ambainis, A.: Quantum walks and their algorithmic applications. Int. J. Quantum Inf. 01(04), 507–518 (2003).  https://doi.org/10.1142/S0219749903000383CrossRefzbMATHGoogle Scholar
  2. 2.
    Ambainis, A.: Quantum walk algorithm for element distinctness. SIAM J. Comput. 37(1), 210–239 (2007).  https://doi.org/10.1137/S0097539705447311MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambainis, A., Kokainis, M.: Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games. In: Proceedings of the 49th STOC. ACM (2017).  https://doi.org/10.1145/3055399.3055444
  4. 4.
    Aono, Y., Nguyen, P.Q., Shen, Y.: Quantum lattice enumeration and tweaking discrete pruning. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11272, pp. 405–434. Springer, Cham (2018).  https://doi.org/10.1007/978-3-030-03326-2_14CrossRefGoogle Scholar
  5. 5.
    Belovs, A., Childs, A.M., Jeffery, S., Kothari, R., Magniez, F.: Time-efficient quantum walks for 3-distinctness. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 105–122. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39206-1_10CrossRefGoogle Scholar
  6. 6.
    Campbell, E., Khurana, A., Montanaro, A.: Applying quantum algorithms to constraint satisfaction problems. Quantum 3, 167 (2018).  https://doi.org/10.22331/q-2019-07-18-167CrossRefGoogle Scholar
  7. 7.
    Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the 35th STOC. ACM (2003).  https://doi.org/10.1145/780542.780552
  8. 8.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962).  https://doi.org/10.1145/368273.368557MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. JACM 7(3), 201–215 (1960).  https://doi.org/10.1145/321033.321034MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24605-3_37CrossRefGoogle Scholar
  11. 11.
    Gomes, C.P., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability solvers. In: Handbook of Knowledge Representation. Elsevier (2008).  https://doi.org/10.1016/S1574-6526(07)03002-7CrossRefGoogle Scholar
  12. 12.
    Gu, J., Purdom, P.W., Franco, J., Wah, B.W.: Algorithms for the satisfiability (SAT) problem: a survey. In: Handbook of Combinatorial Optimization. Springer (1999)  https://doi.org/10.1007/978-1-4757-3023-4_7CrossRefGoogle Scholar
  13. 13.
    Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44(4), 307–327 (2003).  https://doi.org/10.1080/00107151031000110776CrossRefGoogle Scholar
  14. 14.
    Magniez, F., Nayak, A., Roland, J., Santha, M.: Search via quantum walk. In: Proceedings of the 39th STOC. Theory of Computing (2007).  https://doi.org/10.1145/1250790.1250874
  15. 15.
    Malaguti, E., Toth, P.: A survey on vertex coloring problems. Int. Trans. Oper. Res. 17(1), 1–34 (2010).  https://doi.org/10.1111/j.1475-3995.2009.00696.xMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Montanaro, A.: Quantum walk speedup of backtracking algorithms. Theory Comput. 14(15), 1–24 (2018).  https://doi.org/10.4086/toc.2018.v014a015MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Montanaro, A.: Quantum speedup of branch-and-bound algorithms. arXiv:1906.10375 (2019)
  18. 18.
    Montanaro, A.: Data from Quantum algorithms for CSPs. c9pb. Accessed Jul 2019.  https://doi.org/10.5523/bris.19va21gun3c7629f291kmd6w37
  19. 19.
    Santha, M.: Quantum walk based search algorithms. In: Agrawal, M., Du, D., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 31–46. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-79228-4_3CrossRefGoogle Scholar
  20. 20.
    Szegedy, M.: Quantum speed-up of Markov chain based algorithms. In: FOCS 2004. IEEE (2004).  https://doi.org/10.1109/FOCS.2004.53
  21. 21.
    van Beek, P.: Backtracking search algorithms. In: Handbook of Constraint Programming. Elsevier (2006).  https://doi.org/10.1016/S1574-6526(06)80008-8CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Atos, Quantum R&DLes Clayes-sous-BoisFrance

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