Dimension reduction and redundancy removal through successive Schmidt decompositions
arXiv:2302.04801 · doi:10.3390/app13053172
Abstract
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.
References in corpus (22)
- Quantum Machine Learning
- Quantum Mechanics helps in searching for a needle in a haystack
- Quantum Convolutional Neural Networks
- Quantum principal component analysis
- An introduction to quantum machine learning
- Quantum algorithms: an overview
- Simulating Hamiltonian dynamics with a truncated Taylor series
- Power of data in quantum machine learning
- A Schmidt number for density matrices
- Even more efficient quantum computations of chemistry through tensor hypercontraction
- A quantum-inspired classical algorithm for recommendation systems
- Unbiasing Fermionic Quantum Monte Carlo with a Quantum Computer
- Learning a local Hamiltonian from local measurements
- Doubling the size of quantum simulators by entanglement forging
- Determining a local Hamiltonian from a single eigenstate
- Tensor hypercontraction: A universal technique for the resolution of matrix elements of local, finite-range -body potentials in many-body quantum problems
- Maximal entropy approach for quantum state tomography
- Universal Programmable Quantum Circuit Schemes to Emulate an Operator
- Stochastic Estimation of Dynamical Variables
- Convergence of reconstructed density matrix to a pure state using maximal entropy approach
- Variational Approach to Quantum State Tomography based on Maximal Entropy Formalism
- Context aware quantum simulation of a matrix stored in quantum memory