  Isomap algorithm is a representative manifold learning algorithm. The
algorithm simplifies the data analysis process and is widely used in
neuroimaging, spectral analysis and other fields. However, the classic Isomap
algorithm becomes unwieldy when dealing with large data sets. Our object is to
accelerate the classical algorithm with quantum computing, and propose the
quantum Isomap algorithm. The algorithm consists of two sub-algorithms. The
first one is the quantum Floyd algorithm, which calculates the shortest
distance for any two nodes. The other is quantum Isomap algorithm based on
quantum Floyd algorithm, which finds a low-dimensional representation for the
original high-dimensional data. Finally, we analyze that the quantum Floyd
algorithm achieves exponential speedup without sampling. In addition, the time
complexity of quantum Isomap algorithm is $O(dNpolylogN)$. Both algorithms
reduce the time complexity of classical algorithms.
