Dimension reduction and redundancy removal through successive Schmidt decompositions
This work addresses the need for efficient data mapping and simulation in quantum computing, offering a method that is incremental but applicable to both classical data processing and quantum algorithm verification.
The paper tackles the problem of efficiently mapping classical data onto quantum states and approximating quantum operations for classical simulation by using successive Schmidt decompositions to approximate matrices and vectors with tensor products. It shows that data with distributions like uniform or Poisson and quantum operations like the quantum Fourier transform can be approximated with only a few terms, enabling easier quantum circuit mapping or classical simulation, as demonstrated on datasets such as iris flower and 20newsgroup and quantum Hamiltonians like the transverse field Ising model.
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.