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Lecture notes in networks and systems

Matrix Product State Representations for Machine Learning

Eric M. Howard, Iftekher S. Chowdhury, Ian Nagle

Published Jan 1, 2021• 1 citation
Publisher
DOI
10.1007/978-3-030-77445-5_43

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ABSTRACT The exact solution of strongly correlated systems is a computationally demanding problem, scaling exponentially with system size. Over the decades, several approximate methods have been developed to address this scaling problem. Among these, matrix product state (MPS) has emerged as physically intuitive and polynomially scaling. Methods such as density matrix renormalization group (DMRG), time evolution block decimation (TEBD), etc., use the underlying MPS wavefunction ansatz to optimize it variationally. MPS can also be utilized as a machine learning model to select the most important electronic configurations, as employed in selected configuration interactions. This work uses machine learning (ML) optimization of MPS wavefunctions to eliminate the need for multiple diagonalizations during the optimization process. We compare the performance of the machine‐learned MPS‐assisted selected configuration interaction approach, often mentioned as a classifier, to the machine‐learned MPS wavefunction ansatz itself. While showing the viability of the machine‐learned MPS ansatz, it is observed that within ML, MPS is best used as a selected configuration interaction.
Machine learning (ML) has been used to optimize the matrix product state (MPS) ansatz for the wavefunction of strongly correlated systems. The ML optimization of MPS has been tested for the Heisenberg Hamiltonian on one-dimensional and ladder lattices, which correspond to conjugated molecular systems. The input descriptors and output for the supervised ML are lattice configurations and configuration interaction coefficients, respectively. Efficient learning can be achieved from data over the full Hilbert space via exact diagonalization or full configuration interaction, as well as over a much smaller sub-space via Monte Carlo Configuration Interaction. We show that this circumvents the need to calculate energy and operator expectation values and is therefore a computationally efficient alternative to variational optimization.