Machine Learns Quantum Complexity

TL;DR

This paper demonstrates that deep learning algorithms can effectively learn and predict Krylov spread complexity in quantum systems, capturing essential features of quantum states across all timescales, with performance depending on basis choice.

Contribution

It introduces a neural network approach to learn quantum Krylov spread complexity, highlighting basis dependence and late-time behavior in quantum systems.

Findings

Neural network successfully learns Krylov spread complexity across all timescales.

Performance depends on the basis choice, working well with energy eigenbasis or Krylov basis.

Deep learning captures temperature-dependent features and late-time features of quantum states.

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.