We study the problem of learning an unknown graph provided via an oracle using a quantum algorithm. We consider three query models. In the first model ("OR queries"), the oracle returns whether a given subset of the vertices contains any edges. In the second ("parity queries"), the oracle returns the parity of the number of edges in a subset. In the third model, we are given copies of the graph state corresponding to the graph.
We give quantum algorithms that achieve speedups over the best possible classical algorithms in the OR and parity query models, for some families of graphs, and give quantum algorithms in the graph state model whose complexity is similar to the parity query model. For some parameter regimes, the speedups can be exponential in the parity query model. On the other hand, without any promise on the graph, no speedup is possible in the OR query model.
A main technique we use is the quantum algorithm for solving the combinatorial group testing problem, for which a query-efficient quantum algorithm was given by Belovs. Here we additionally give a time-efficient quantum algorithm for this problem, based on the algorithm of Ambainis et al.\ for a "gapped" version of the group testing problem. We also give simple time-efficient quantum algorithms based on Fourier sampling and amplitude amplification for learning the exact-half and majority functions, which almost match the optimal complexity of Belovs' algorithms.
Beyond Robertson-Schrödinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs
Graduate School of Information Sciences, Tohoku University（东北大学信息科学研究院）；College of Systems Engineering and Science, Shibaura Institute of Technology（芝浦工业大学系统工程科学学部）；Department of Mathematics, Faculty of Engineering, Shinshu University（信州大学工学部数学系）；Institute of Industrial Science, The University of Tokyo（东京大学工业技术研究所）；Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University（尼古拉·哥白尼大学物理、天文学与信息学学院物理研究所）
