Improved quantum backtracking algorithms using effective resistance estimates

We investigate quantum backtracking algorithms of a type previously introduced by Montanaro (arXiv:1509.02374). These algorithms explore trees of unknown structure, and in certain cases exponentially outperform classical procedures (such as DPLL). Some of the previous work focused on obtaining a quantum advantage for trees in which a unique marked vertex is promised to exist. We remove this restriction and re-characterise the problem in terms of the effective resistance of the search space. To this end, we present a generalisation of one of Montanaro's algorithms to trees containing $k \geq 1$ marked vertices, where $k$ is not necessarily known \textit{a priori}. Our approach involves using amplitude estimation to determine a near-optimal weighting of a diffusion operator, which can then be applied to prepare a superposition state that has support only on marked vertices and ancestors thereof. By repeatedly sampling this state and updating the input vertex, a marked vertex is reached in a logarithmic number of steps. The algorithm thereby achieves the conjectured bound of $\widetilde{\mathcal{O}}(\sqrt{TR_{\mathrm{max}}})$ for finding a single marked vertex and $\widetilde{\mathcal{O}}\left(k\sqrt{T R_{\mathrm{max}}}\right)$ for finding all $k$ marked vertices, where $T$ is an upper bound on the tree size and $R_{\mathrm{max}}$ is the maximum effective resistance encountered by the algorithm. This constitutes a speedup over Montanaro's original procedure in both the case of finding one and finding multiple marked vertices in an arbitrary tree. If there are no marked vertices, the effective resistance becomes infinite, and we recover the scaling of Montanaro's existence algorithm.