Abstract
Combining insights from machine learning and quantum Monte Carlo, the stochastic reconfiguration method with neural network Ansatz states is a promising new direction for high-precision ground state estimation of quantum many-body problems. Even though this method works well in practice, little is known about the learning dynamics. In this paper, we bring to light several hidden details of the algorithm by analyzing the learning landscape. In particular, the spectrum of the quantum Fisher matrix of complex restricted Boltzmann machine states exhibits a universal initial dynamics, but the converged spectrum can dramatically change across a phase transition. In contrast to the spectral properties of the quantum Fisher matrix, the actual weights of the network at convergence do not reveal much information about the system or the dynamics. Furthermore, we identify a new measure of correlation in the state by analyzing entanglement in eigenvectors. We show that, generically, the learning landscape modes with least entanglement have largest eigenvalue, suggesting that correlations are encoded in large flat valleys of the learning landscape, favoring stable representations of the ground state.
Highlights
The fields of machine learning and quantum information science have seen a lot of crossbreeding
We analyze the spectral properties of the quantum Fisher matrix during the learning process of finding the ground state of the transverse field Ising (TFI) model
The spectral properties of the quantum Fisher matrix as well as the energy during the learning process obtained from the simulation are plotted in Fig. 2
Summary
The fields of machine learning and quantum information science have seen a lot of crossbreeding. The variational quantum eigensolver [2]—perhaps the most promising quantum algorithms for first-generation quantum computers—is based on the variational optimization of a cost function to be evaluated on a quantum device, providing a new playground for hybrid quantum-classical learning [3,4]. A number of studies have shown that machinelearning-inspired sampling algorithms can reach state-of-theart precision, including ground-state energy estimation [5,6,7,8], time evolution [5,9], identifying phase transitions [10,11,12], and decoding quantum error correcting codes [13,14] The authors show that ground-state energy evaluations can outperform the state-ofthe-art tensor network methods on benchmark problems
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