Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

2025-07-01

AI Summary

This work addresses the quantum acceleration of dynamic programming (DP) algorithms. We propose the first general-purpose quantum framework for polynomial-time DP. Methodologically, leveraging the average degree δ of the DP dependency graph, we integrate quantum query access with recursive structure, exploiting quantum superposition and interference to accelerate subproblem evaluation. The resulting algorithm achieves a time complexity of $ ilde{O}(|V|sqrt{delta})$, while retaining the same space complexity as classical DP. Our key contribution is the establishment of the first systematic quantum DP paradigm; based on it, we design a quantum Bellman–Ford algorithm with time complexity $ ilde{O}(nsqrt{nm})$. This algorithm outperforms the best-known classical counterpart when $m in Omega(n^{1.4})$, marking a breakthrough in quantum DP that simultaneously achieves generality and practicality.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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 $δ$ 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 $ ilde{O}(|V(G_{mathcal{P}}(I))| sqrtδ)$ 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 $ ilde{O}(nsqrt{nm})$ time, which improves the best known classical upper bound when $m in Ω(n^{1.4})$.

Problem

• Extends classical dynamic programming to quantum algorithms
• Retains classical space complexity with computational speedup
• Solves problems in O(|V|√δ) time using dependency graphs

Innovation

• Quantum dynamic programming framework extension
• Same space complexity, computational speedup
• Bellman-Ford quantum version improves runtime