Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

• URL: http://arxiv.org/abs/2507.00823v1
• Date: Tue, 01 Jul 2025 14:55:18 GMT
• Title: Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms
• Authors: Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista, Michael T. Goodrich, Martin Nöllenburg,
• Abstract summary: We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms.<n>The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup.
• Score: 4.832760917132771
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem $\mathcal P$ and an instance $I$ of $\mathcal P$, such a speedup can be expressed in terms of the average degree $\delta$ of the dependency digraph $G_{\mathcal{P}}(I)$ of $I$, determined by a recursive formulation of $\mathcal P$. The nodes of this graph are the subproblems of $\mathcal P$ induced by $I$ and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in $\tilde{O}(|V(G_{\mathcal{P}}(I))| \sqrt{\delta})$ time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted $n$-vertex digraph with $m$ edges that runs in $\tilde{O}(n\sqrt{nm})$ time, which improves the best known classical upper bound when $m \in \Omega(n^{1.4})$.

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