A Quantum Algorithm for Minimum Steiner Tree Problem.

Minimum Steiner tree problem is a well-known NP-hard problem. For the minimum Steiner tree problem in graphs with $n$ vertices and $k$ terminals, there are many classical algorithms that take exponential time in $k$. In this paper, to the best of our knowledge, we propose the first quantum algorithm for the minimum Steiner tree problem. The complexity of our algorithm is $\mathcal{O}^*(1.812^k)$. A key to realize the proposed method is how to reduce the computational time of dynamic programming by using a quantum algorithm because existing classical (non-quantum) algorithms in the problem rely on dynamic programming. Fortunately, dynamic programming is realized by a quantum algorithm for the travelling salesman problem, in which Grover's quantum search algorithm is introduced. However, due to difference between their problem and our problem to be solved, recursions are different. Hence, we cannot apply their technique to the minimum Steiner tree problem in that shape. We solve this issue by introducing a decomposition of a graph proposed by Fuchs et al.