Random-State Generation and Preparation Complexity in Rydberg Atom Arrays

Rydberg atom arrays are powerful platforms for studying quantum many-body systems. We consider the Rydberg-Ising Hamiltonian on periodic chains and numerically study ensembles of states generated by random global pulse sequences subject to hardware constraints and fixed evolution times. We compare the statistical properties of such states with those of Haar-random states within the relevant lattice symmetry sector. In the strong-interaction regime (short interatomic distance), the dynamics is governed by an effective blockade that restricts Hilbert-space exploration and limits entanglement growth. In this regime, level-spacing statistics of reduced density matrices are close to random-matrix predictions, while the distribution of measurement probabilities deviates from Porter-Thomas behavior. For weaker interactions (larger interatomic distance), the system approaches Haar-like statistics at long times, as reflected in entanglement entropy, entanglement spectrum statistics, and the distribution of measurement probabilities. At intermediate interactions, this behavior is observed on experimentally relevant timescales. Motivated by this observation, we investigate whether generic symmetric quantum states can be efficiently prepared using quantum optimal control in this regime. Employing target states drawn from an ensemble with a broad entropy distribution, we observe high fidelities (infidelities between $10^{-5}$ and $3\times 10^{-2}$ for 9 spins). The fidelity, however, decreases with the entanglement entropy of the target state, demonstrating that highly entangled states are intrinsically harder to prepare under realistic constraints.