Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs

Authors Adam Glos , Martins Kokainis , Ryuhei Mori , Jevgēnijs Vihrovs

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Adam Glos
  • Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Warsaw, Poland
Martins Kokainis
  • Centre for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia
Ryuhei Mori
  • School of Computing, Tokyo Institute of Technology, Japan
Jevgēnijs Vihrovs
  • Center for Quantum Computer Science, Faculty of Computing, University of Latvia, Riga, Latvia


We would like to thank Krišjānis Prūsis for helpful discussions and comments. We also thank anonymous reviewers for helpful comments and suggestions on the presentation.

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Adam Glos, Martins Kokainis, Ryuhei Mori, and Jevgēnijs Vihrovs. Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 50:1-50:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the n-dimensional lattice graph Q(D,n) with vertices in {0,1,…,D}ⁿ. We study the complexity of the following problem: given a subgraph G of Q(D,n) via query access to the edges, determine whether there is a path from 0ⁿ to Dⁿ. While the classical query complexity is Θ̃((D+1)ⁿ), we show a quantum algorithm with complexity Õ(T_Dⁿ), where T_D < D+1. The first few values of T_D are T₁ ≈ 1.817, T₂ ≈ 2.660, T₃ ≈ 3.529, T₄ ≈ 4.421, T₅ ≈ 5.332. We also prove that T_D ≥ (D+1)/e (here, e ≈ 2.718 is the Euler’s number), thus for general D, this algorithm does not provide, for example, a speedup, polynomial in the size of the lattice. While the presented quantum algorithm is a natural generalization of the known quantum algorithm for D = 1 by Ambainis et al., the analysis of complexity is rather complicated. For the precise analysis, we use the saddle-point method, which is a common tool in analytic combinatorics, but has not been widely used in this field. We then show an implementation of this algorithm with time and space complexity poly(n)^{log n} T_Dⁿ in the QRAM model, and apply it to the Set Multicover problem. In this problem, m subsets of [n] are given, and the task is to find the smallest number of these subsets that cover each element of [n] at least D times. While the time complexity of the best known classical algorithm is O(m(D+1)ⁿ), the time complexity of our quantum algorithm is poly(m,n)^{log n} T_Dⁿ.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum query complexity
  • Theory of computation → Dynamic programming
  • Quantum query complexity
  • Dynamic programming
  • Lattice graphs


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  1. Amir Abboud. Fine-Grained Reductions and Quantum Speedups for Dynamic Programming. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), volume 132 of Leibniz International Proceedings in Informatics (LIPIcs), pages 8:1-8:13, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.8.
  2. Daniel S. Abrams and Seth Lloyd. Simulation of many-body fermi systems on a universal quantum computer. Phys. Rev. Lett., 79:2586-2589, September 1997. URL: https://doi.org/10.1103/PhysRevLett.79.2586.
  3. Andris Ambainis. Quantum search with variable times. Theory of Computing Systems, 47(3):786-807, 2010. URL: https://doi.org/10.1007/s00224-009-9219-1.
  4. Andris Ambainis, Kaspars Balodis, Jānis Iraids, Martins Kokainis, Krišjānis Prūsis, and Jevgēnijs Vihrovs. Quantum speedups for exponential-time dynamic programming algorithms. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '19, page 1783–1793, USA, 2019. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611975482.107.
  5. Richard Bellman. Dynamic programming treatment of the travelling salesman problem. J. ACM, 9(1):61-63, 1962. URL: https://doi.org/10.1145/321105.321111.
  6. Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani. Strengths and weaknesses of quantum computing. SIAM Journal on Computing, 26(5):1510-1523, 1997. URL: https://doi.org/10.1137/S0097539796300933.
  7. Hans L. Bodlaender, Fedor V. Fomin, Arie M. C. A. Koster, Dieter Kratsch, and Dimitrios M. Thilikos. A note on exact algorithms for vertex ordering problems on graphs. Theory of Computing Systems, 50(3):420-432, 2012. URL: https://doi.org/10.1007/s00224-011-9312-0.
  8. Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. URL: https://doi.org/10.1016/S0304-3975(01)00144-X.
  9. David Burshtein and Gadi Miller. Asymptotic enumeration methods for analyzing LDPC codes. IEEE Transactions on Information Theory, 50:1115-1131, 2004. URL: https://doi.org/10.1109/TIT.2004.828064.
  10. Mitchell Chiew, Kooper de Lacy, Chao-Hua Yu, Sam Marsh, and Jingbo B. Wang. Graph comparison via nonlinear quantum search. Quantum Information Processing, 18:302, August 2019. URL: https://doi.org/10.1007/s11128-019-2407-2.
  11. Arjan Cornelissen, Stacey Jeffery, Maris Ozols, and Alvaro Piedrafita. Span Programs and Quantum Time Complexity. In Javier Esparza and Daniel Kráľ, editors, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020), volume 170 of Leibniz International Proceedings in Informatics (LIPIcs), pages 26:1-26:14, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.26.
  12. Marek Cygan and Marcin Pilipczuk. Faster exact bandwidth. In Graph-Theoretic Concepts in Computer Science, pages 101-109, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-540-92248-3_10.
  13. N. G. de Bruijn, C^A. van Ebbenhorst Tengbergen, and D. Kruyswijk. On the set of divisors of a number. Nieuw Archief voor Wiskunde, 23(2):191-193, 1951. URL: https://research.tue.nl/en/publications/on-the-set-of-divisors-of-a-number.
  14. Christoph Dürr and Peter Høyer. A quantum algorithm for finding the minimum, 1996. URL: http://arxiv.org/abs/quant-ph/9607014.
  15. Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, USA, 1 edition, 2009. URL: https://doi.org/10.1017/CBO9780511801655.
  16. Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. Springer Science & Business Media, 2010. Google Scholar
  17. Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum random access memory. Phys. Rev. Lett., 100:160501, 2008. URL: https://doi.org/10.1103/PhysRevLett.100.160501.
  18. Adam Glos, Martins Kokainis, Ryuhei Mori, and Jevgēnijs Vihrovs. Quantum speedups for dynamic programming on n-dimensional lattice graphs, 2021. URL: http://arxiv.org/abs/2104.14384.
  19. I. J. Good. Saddle-point Methods for the Multinomial Distribution. The Annals of Mathematical Statistics, 28(4):861-881, 1957. URL: https://doi.org/10.1214/aoms/1177706790.
  20. Joaquim A. S. Gromicho, Jelke J. van Hoorn, Francisco Saldanha da Gama, and Gerrit T. Timmer. Solving the job-shop scheduling problem optimally by dynamic programming. Computers & Operations Research, 39(12):2968-2977, 2012. URL: https://doi.org/10.1016/j.cor.2012.02.024.
  21. Lov K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC '96, page 212–219, New York, NY, USA, 1996. Association for Computing Machinery. URL: https://doi.org/10.1145/237814.237866.
  22. Michael Held and Richard M. Karp. A dynamic programming approach to sequencing problems. Journal of SIAM, 10(1):196-210, 1962. URL: https://doi.org/10.1145/800029.808532.
  23. Peter Høyer, Michele Mosca, and Ronald de Wolf. Quantum search on bounded-error inputs. In Automata, Languages and Programming, ICALP'03, page 291–299, Berlin, Heidelberg, 2003. Springer-Verlag. URL: https://doi.org/10.1007/3-540-45061-0_25.
  24. Qiang-Sheng Hua, Yuexuan Wang, Dongxiao Yu, and Francis C.M. Lau. Dynamic programming based algorithms for set multicover and multiset multicover problems. Theoretical Computer Science, 411(26):2467-2474, 2010. URL: https://doi.org/10.1016/j.tcs.2010.02.016.
  25. Bronisław Knaster, Casimir Kuratowski, and Stefan Mazurkiewicz. Ein beweis des fixpunktsatzes für n-dimensionale simplexe. Fundamenta Mathematicae, 14(1):132-137, 1929. URL: https://doi.org/10.4064/fm-14-1-132-137.
  26. Masayuki Miyamoto, Masakazu Iwamura, Koichi Kise, and François Le Gall. Quantum speedup for the minimum steiner tree problem. In Donghyun Kim, R. N. Uma, Zhipeng Cai, and Dong Hoon Lee, editors, Computing and Combinatorics, pages 234-245, Cham, 2020. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-58150-3_19.
  27. Jesper Nederlof. Inclusion exclusion for hard problems. Master’s thesis, Utrecht University, 2008. URL: https://webspace.science.uu.nl/~neder003/MScThesis.pdf.
  28. Harilaos N. Psaraftis. A dynamic programming approach for sequencing groups of identical jobs. Operations Research, 28(6):1347-1359, 1980. URL: https://doi.org/10.1287/opre.28.6.1347.
  29. Kazuya Shimizu and Ryuhei Mori. Exponential-time quantum algorithms for graph coloring problems. In Yoshiharu Kohayakawa and Flávio Keidi Miyazawa, editors, LATIN 2020: Theoretical Informatics, pages 387-398, Cham, 2020. Springer International Publishing. URL: http://arxiv.org/abs/1907.00529.
  30. Seiichiro Tani. Quantum Algorithm for Finding the Optimal Variable Ordering for Binary Decision Diagrams. In Susanne Albers, editor, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020), volume 162 of Leibniz International Proceedings in Informatics (LIPIcs), pages 36:1-36:19, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.SWAT.2020.36.
  31. Jelke J. van Hoorn, Agustín Nogueira, Ignacio Ojea, and Joaquim A. S. Gromicho. An corrigendum on the paper: Solving the job-shop scheduling problem optimally by dynamic programming. Computers & Operations Research, 78:381, 2017. URL: https://doi.org/10.1016/j.cor.2016.09.001.
  32. Łukasz Kowalik, Shaohua Li, Wojciech Nadara, Marcin Smulewicz, and Magnus Wahlström. Many Visits TSP Revisited. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, 28th Annual European Symposium on Algorithms (ESA 2020), volume 173 of Leibniz International Proceedings in Informatics (LIPIcs), pages 66:1-66:22, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.66.
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