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Quantum Speedups for Dynamic Programming on n-Dimensional Lattice Graphs

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



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Author Details

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

Acknowledgements

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.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.MFCS.2021.50

Abstract

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
Keywords
  • Quantum query complexity
  • Dynamic programming
  • Lattice graphs

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