An Empirical Study of Quantum Dynamics as a Ground State Problem with Neural Quantum States

TL;DR

This paper investigates the use of neural quantum states to encode the ground state of a quantum spin chain's time evolution, highlighting their expressivity and the challenges in training as system complexity grows.

Contribution

It provides an empirical assessment of neural quantum states' ability to approximate ground states in the Feynman-Kitaev formalism, focusing on expressivity and trainability challenges.

Findings

Neural quantum states can accurately approximate the ground state.

Training difficulty increases with more time steps due to entanglement.

Hyperparameter tuning is crucial for successful training.

Abstract

We consider the Feynman-Kitaev formalism applied to a spin chain described by the transverse field Ising model. This formalism consists of building a Hamiltonian whose ground state encodes the time evolution of the spin chain at discrete time steps. To find this ground state, variational wave functions parameterised by artificial neural networks -- also known as neural quantum states (NQSs) -- are used. Our work focuses on assessing, in the context of the Feynman-Kitaev formalism, two properties of NQSs: expressivity (the possibility that variational parameters can be set to values such that the NQS is faithful to the true ground state of the system) and trainability (the process of reaching said values). We find that the considered NQSs are capable of accurately approximating the true ground state of the system, i.e., they are expressive enough ans\"atze. However, extensive hyperparameter tuning experiments show that, empirically, reaching the set of values for the variational parameters that correctly describe the ground state becomes ever more difficult as the number of time steps increase because the true ground state becomes more entangled, and the probability distribution starts to spread across the Hilbert space canonical basis.