The Problem of Dynamic Programming on a Quantum Computer

We discuss the problem of finite-horizon dynamic programming (DP) on a quantum computer. We introduce a query model for studying quantum and classical algorithms for solving DP problems, and provide example oracle constructions for the travelling salesperson problem, the minimum set-cover problem, and the edit distance problem. We formulate open questions regarding quadratic quantum speedups for DP and discuss their implications. We then prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for DP, and show that no greater-than-quadratic speedup can be achieved for solving DP problems.
Comment: We were not able to amend the proof of Theorem III.5 of the previous version (v2) of this manuscript. Therefore, we have hereby withdrawn the query complexity upper bound claims that were stated in our earlier submission