We study the quantum dynamics of the encoding scheme proposed in [Nguyen et al., PRX Quantum 4, 010316 (2023)], which encodes optimization problems on graphs with arbitrary connectivity into Rydberg atom arrays. Here, a graph vertex is represented by a wire of atoms, and the (crossing) crossing-with-edge gadget is placed at the intersection of two wires to (de)couple their degrees of freedom and reproduce the graph connectivity. We consider the fundamental geometry of two vertex-wires intersecting via a single gadget and look at minimum gap scaling with system size along adiabatic protocols. We find that both polynomial and exponential scaling are possible and, by means of perturbation theory, we relate the exponential closing of the minimum gap to an unfavorable localization of the ground-state wavefunction. Then, on the QuEra Aquila neutral atom machine, we observe such localization and its effect on the success probability of finding the correct solution to the encoded optimization problem. Finally, we propose possible strategies to avoid this quantum bottleneck, leading to an exponential improvement in the adiabatic performance.

We present a biased atomic qubit, universally implementable across all atomic platforms, encoded as a `spin-cat' within ground state Zeeman levels. The key characteristic of our configuration is the coupling of the ground state spin manifold of size Fg≫1 to an excited Zeeman spin manifold of size Fe=Fg−1 using light. This coupling results in eigenstates of the driven atom that include exactly two dark states in the ground state manifold, which are decoupled from light and immune to spontaneous emission from the excited states. These dark states constitute the `spin-cat', leading to the designation `dark spin-cat'. We demonstrate that under strong Rabi drive and for large Fg, the `dark spin-cat' is autonomously stabilized against common noise sources and encodes a qubit with significantly biased noise. Specifically, the bit-flip error rate decreases exponentially with Fg relative to the dephasing rate. We provide an analysis of dark spin-cats, their robustness to noise, and discuss bias-preserving single qubit and entangling gates, exemplified on a Rydberg tweezer platform.