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 $T = \sum_v t_v^2$. 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. As a corollary, this gives a quantum algorithm for variable time search with unknown times with optimal query complexity $\text{O}(\sqrt{T \log \min(n,t_{\max})})$, where $T = \sum_i t_i^2$ and $t_{\max} = \max_i t_i$ if $t_i$ is the number of steps required to compute the $i$-th variable. This resolves the open question of the query complexity of variable time search, as the matching lower bound was recently shown by Ambainis, Kokainis and Vihrovs (TQC'23). As another result, we obtain an $\widetilde{\text{O}}(n)$ time algorithm for the geometric task of determining if any three lines among $n$ given intersect at the same point, improving the $\text{O}(n^{1+\text{o}(1)})$ algorithm of Ambainis and Larka (TQC'20).