Machine Learns Quantum Complexity

Abstract

We study how a machine based on deep learning algorithms learns Krylov spread complexity in quantum systems with N x N random Hamiltonians drawn from the Gaussian unitary ensemble. Using thermofield double states as initial conditions, we demonstrate that a convolutional neural network-based algorithm successfully learns the Krylov spread complexity across all timescales, including the late-time plateaus where states appear nearly featureless and random. Performance strongly depends on the basis choice, performing well with the energy eigenbasis or the Krylov basis but failing in the original basis of the random Hamiltonian. The algorithm also effectively distinguishes temperature-dependent features of thermofield double states. Furthermore, we show that the system time variable of state predicted by deep learning is an irrelevant quantity, reinforcing that the Krylov spread complexity well captures the essential features of the quantum state, even at late times.