  Quantum computers are believed to have the ability to process huge data sizes
which can be seen in machine learning applications. In these applications, the
data in general is classical. Therefore, to process them on a quantum computer,
there is a need for efficient methods which can be used to map classical data
on quantum states in a concise manner. On the other hand, to verify the results
of quantum computers and study quantum algorithms, we need to be able to
approximate quantum operations into forms that are easier to simulate on
classical computers with some errors.
  Motivated by these needs, in this paper we study the approximation of
matrices and vectors by using their tensor products obtained through successive
Schmidt decompositions. We show that data with distributions such as uniform,
Poisson, exponential, or similar to these distributions can be approximated by
using only a few terms which can be easily mapped onto quantum circuits. The
examples include random data with different distributions, the Gram matrices of
iris flower, handwritten digits, 20newsgroup, and labeled faces in the wild.
And similarly, some quantum operations such as quantum Fourier transform and
variational quantum circuits with a small depth also may be approximated with a
few terms that are easier to simulate on classical computers. Furthermore, we
show how the method can be used to simplify quantum Hamiltonians: In
particular, we show the application to randomly generated transverse field
Ising model Hamiltonians. The reduced Hamiltonians can be mapped into quantum
circuits easily and therefore can be simulated more efficiently.
