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

TL;DR

This work investigates solving the quantum dynamics of a transverse-field Ising chain by mapping to a Feynman-Kitaev history-state ground state and approximating it with neural quantum states under variational Monte Carlo. It demonstrates that NQSs are expressive enough to capture highly entangled history states, but trainability deteriorates as the clock dimension grows, due to entanglement growth and broad probability support over the Hilbert-space basis. The study compares RBM, modulus-phase RBM, and autoregressive NQSs, showing that MP-RBM can reach very low infidelity (≈$2\times10^{-3}$) for moderate clock sizes, while optimization challenges persist for larger clocks and autoregressive variants. The findings indicate that optimization, not representation, is the main bottleneck for learning FK ground states in this setting, with implications for variational approaches to quantum dynamics. Adiabatic clock-on strategies mitigate some difficulties, suggesting practical routes for scalable NQS-based simulations of quantum dynamics.

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 ansätze. 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.

Figures (4)

• Figure 1: Representation of the physical spin chain enlargement with a clock state (a); and properties of the ground state of the enlarged Hamiltonian in \ref{['eq:bigHamiltonian']} found with exact diagonalisation. (b) shows the second Rényi entropy per spin of a sub-chain of physical spins and (c) shows the ratio of the canonical basis of the Hilbert space that is needed to explain 99% of the probability of the ground state. The main text (\ref{['sec:results']}) explains these plots in-depth.
• Figure 2: Time evolution approximated with an RBM as an NQS. (a)-(d) show the expected value of the average magnetisation $\expval{\hat{\sigma}^z}=\frac{1}{N_S}\sum_{i=1}^{N_S}\expval{\hat{\sigma}_i^z}$. In each panel, curves are shown for the average magnetisation obtained through exact diagonalisation (ground state), estimation of the variational magnetisation with a sample $\mathcal{M}$ (RBM) and exact variational magnetisation (exact RBM), which results from using the complete state vector instead of a sample. The shaded region indicates the estimated fluctuations of magnetisation using the sample $\mathcal{M}$. The lines serve as a guide for the eye only. (e) shows a box plot of infidelity ($1-\abs{\braket{\Phi(N_S,N_T)}{\Psi^{\text{RBM}}_{\vec{\theta^\star}}}}^2$) for the best 10 hyperparameter experiments, where $\vec{\theta}^\star$ indicates that parameters have been optimised until convergence.
• Figure 3: Probability of each element of the canonical basis of $\mathscr{H}$ and time evolution of magnetisation for an MP-RBM ansatz. The top panel shows the $2^9$ probabilities associated to each element of the canonical basis of $\mathscr{H}$ for the ground states obtained through exact diagonalisation, through variational minimisation of the infidelity, and through variational minimisation of the estimated energy in the left, middle and right sub-panels, respectively. The bottom panel shows the average magnetisation obtained with each of these states, where the "ground state" line corresponds to the magnetisation obtained with exact diagonalisation, the "MP-RBM" line is obtained through variational minimisation of the estimated energy, and the "infidelity MP-RBM" is obtained through variational minimisation of the infidelity.
• Figure 4: Similar to \ref{['fig:RBMTimeEvolution']} but for the autoregressive ansatz in \ref{['eq:arnn']}.