We present a general quantum algorithm for solving finite-horizon dynamic programming problems. Up to polylogarithmic factors, our algorithm provides a quadratic quantum advantage in terms of the number of states of a given dynamic programming problem. This speedup comes at the expense of the appearance of other polynomial factors representative of the number of actions of the dynamic programming problem, the maximum value of the instantaneous reward, and the time horizon of the problem. Our algorithm can be applied to combinatorial optimization problems solved classically using dynamic programming techniques. As one application, we show that the travelling salesperson problem can be solved in $O^*(\lceil c \rceil^4 \sqrt{2^n})$ on a quantum computer, where $n$ is the number of vertices of the underlying graph and $\lceil c \rceil$ is its maximum edge-weight. As another example, we show that the minimum set-cover problem can be solved in $O(\sqrt{2^n} \operatorname{poly}(m, n))$, where $m$ is the number of sets used to cover a universe of size $n$. Finally, we prove lower bounds for the query complexity of quantum algorithms and classical randomized algorithms for solving dynamic programming problems, and show that no greater-than-quadratic speedup in either the number of states or number of actions can be achieved for solving dynamic programming problems using quantum algorithms.

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