Geometry of learning neural quantum states

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.

arXiv comment18 pages, 10 Figures, close to a version published in Physical Review Research

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• Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape
• Neural Error Mitigation of Near-Term Quantum Simulations
• Characterizing the loss landscape of variational quantum circuits
• Role of stochastic noise and generalization error in the time propagation of neural-network quantum states
• Dynamics with autoregressive neural quantum states: application to critical quench dynamics
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• An Alternative Approach to Quantum Imaginary Time Evolution
• Correlation-Enhanced Neural Networks as Interpretable Variational Quantum States
• Automatic Differentiable Monte Carlo: Theory and Application
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