31,353 research outputs found
The Birkhoff theorem for unitary matrices of arbitrary dimensions
It was shown recently that Birkhoff's theorem for doubly stochastic matrices
can be extended to unitary matrices with equal line sums whenever the dimension
of the matrices is prime. We prove a generalization of the Birkhoff theorem for
unitary matrices with equal line sums for arbitrary dimension.Comment: This manuscript presents a proof for the general unitary birkhoff
theorem, conjectured in arXiv:1509.0862
Quantum Dimensionality Reduction by Linear Discriminant Analysis
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 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 of vectors and a
quadratic speedup on the dimensionality 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
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