Solving the quantum many-body Hamiltonian learning problem with neural differential equations

Abstract Understanding and characterising quantum many-body dynamics remains a significant challenge due to both the exponential complexity required to represent quantum many-body Hamiltonians, and the need to accurately track states in time under the action of such Hamiltonians. This inherent complexity limits our ability to characterise quantum many-body systems, highlighting the need for innovative approaches to unlock their full potential. To address this challenge, we propose a novel method to solve the Hamiltonian learning (HL) problem -inferring quantum dynamics from many-body state trajectories. Our approach has the unprecedented advantage of distilling and filtering an estimator Hamiltonian from a Neural ODE representation of state trajectories. By distilling Hamiltonian information from Neural ODE onto an ansatz Hamiltonian, our method is reliably convergent, experimentally friendly, and interpretable. In addition to this, we propose a new quantitative benchmark based on power laws, which can objectively compare the reliability and generalisation capabilities of any two HL algorithms. Finally, we benchmark our method against state-of-the-art HL algorithms with a 1D spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> chain proof of concept, giving a stable solution for HL on a set of Hamiltonians previously unlearnable in the literature. Code for our method is available and open source.