We study the application of sample-based Krylov quantum diagonalization (SKQD) to a two-dimensional pure $\mathbb{Z}_2$ lattice gauge theory.
SKQD samples configurations
from states obtained via time-evolution quantum circuits, projects the Hamiltonian into the subspace spanned by the sampled configurations, and solves an eigenvalue problem to estimate the ground-state energy.
We consider a model defined on a triangular lattice, which allows for an efficient construction of time-evolution circuits on the IBM Quantum Heron processor. We also propose a configuration recovery method based on Gauss's law and a conditional restricted Boltzmann machine (CRBM), to mitigate the effects of noise.
We simulate a lattice of 108 links using the superconducting quantum processor $\texttt{ibm}\_\texttt{kawasaki}$, which allows us to obtain the ground-state energy within a few percent of tensor network results.
We also compare configuration recovery with and without the CRBM, and observe that the CRBM improves the accuracy at each iteration.

