  Dimensionality reduction (DR) of data is a crucial issue for many machine
learning tasks, such as pattern recognition and data classification. In this
paper, we present a quantum algorithm and a quantum circuit to efficiently
perform linear discriminant analysis (LDA) for dimensionality reduction.
Firstly, the presented algorithm improves the existing quantum LDA algorithm to
avoid the error caused by the irreversibility of the between-class scatter
matrix $S_B$ in the original algorithm. Secondly, a quantum algorithm and
quantum circuits are proposed to obtain the target state corresponding to the
low-dimensional data. Compared with the best-known classical algorithm, the
quantum linear discriminant analysis dimensionality reduction (QLDADR)
algorithm has exponential acceleration on the number $M$ of vectors and a
quadratic speedup on the dimensionality $D$ of the original data space, when
the original dataset is projected onto a polylogarithmic low-dimensional space.
Moreover, the target state obtained by our algorithm can be used as a submodule
of other quantum machine learning tasks. It has practical application value of
make that free from the disaster of dimensionality.
