  Reducing the dimension of nonlinear data is crucial in data processing and
visualization. The locally linear embedding algorithm (LLE) is specifically a
representative nonlinear dimensionality reduction method with well maintaining
the original manifold structure. In this paper, we present two implementations
of the quantum locally linear embedding algorithm (QLLE) to perform the
nonlinear dimensionality reduction on quantum devices. One implementation, the
linear-algebra-based QLLE algorithm, utilizes quantum linear algebra
subroutines to reduce the dimension of the given data. The other
implementation, the variational quantum locally linear embedding algorithm
(VQLLE) utilizes a variational hybrid quantum-classical procedure to acquire
the low-dimensional data. The classical LLE algorithm requires polynomial time
complexity of $N$, where $N$ is the global number of the original
high-dimensional data. Compared with the classical LLE, the
linear-algebra-based QLLE achieves quadratic speedup in the number and
dimension of the given data. The VQLLE can be implemented on the near term
quantum devices in two different designs. In addition, the numerical
experiments are presented to demonstrate that the two implementations in our
work can achieve the procedure of locally linear embedding.
