Quantum Search on Computation Trees

We show a simple generalization of the quantum walk algorithm for search in backtracking trees by Montanaro (ToC 2018) to the case where vertices can have different times of computation. If a vertex \(v\) in the tree of depth \(D\) is computed in \(t_v\) steps from its parent, then we show that detection of a marked vertex requires \(\text O(\sqrt {TD})\) queries to the steps of the computing procedures, where \(T = \sum_v t^2_v\). This framework provides an easy and convenient way to re-obtain a number of other quantum frameworks like variable time search, quantum divide & conquer and bomb query algorithms. The underlying algorithm is simple, explicitly constructed, and has low poly-logarithmic factors in the complexity.