Local Rydberg blockade regimes for disk graph embedding and quantum optimization

TL;DR

This work develops a local Rydberg blockade framework to embed disk graphs into neutral-atom arrays, enabling MIS optimization beyond unit-disk graphs. It introduces two metrics—the correlation matrix and maximum independence violation—to quantify embedding fidelity under inhomogeneous drives, and demonstrates that local driving enables meaningful disk-graph embeddings where global driving fails. Finite local drives improve approximate MIS performance, increasing the probability of near-optimal independent sets on non-unit-disk instances. The approach broadens the computational reach of Rydberg platforms for analog quantum optimization and provides a path toward tackling more complex combinatorial problems.

Abstract

Rydberg atom arrays are a powerful platform for solving combinatorial optimization problems, owing to the Rydberg blockade mechanism, which imposes effective constraints on simultaneous atomic excitations. These constraints have enabled the encoding of the Maximum Independent Set (MIS) problem on unit disk graphs, where atoms interact within a fixed, globally defined blockade radius. However, this restriction limits the class of addressable problems. A natural extension is to consider disk graphs, which generalize unit disk graphs by allowing arbitrary disk radii and correspond to the intersection graphs of disks in the plane. Embedding such graphs in Rydberg systems requires moving beyond the standard, globally uniform blockade model. In this work, we introduce a local Rydberg blockade regime, which emerges when local drives are applied to different pairs of atoms involved in a potential interaction. We develop a general theoretical framework for this regime and propose two novel metrics, the correlation matrix and the maximum independence violation, to quantify the quality of the embedding. Using these metrics, we demonstrate that disk graphs can be meaningfully embedded into Rydberg atom arrays under local drive schemes, thereby expanding the landscape of quantum-addressable optimization problems. Finally, when evaluating approximate solutions of the MIS problem, characterized by near-optimal independent sets, local drive approaches exhibit significantly improved performance over global ones. These results highlight the practical advantage of local blockade engineering for approximate combinatorial optimization and open a path toward leveraging the analog capabilities of Rydberg platforms beyond conventional geometric constraints.

Figures (6)

• Figure 1: (a) The three driving scenarios are illustrated, where the light (dark) green circles correspond to atoms initialized in the ground (excited) state and driven with lasers with amplitudes $\Omega$. (b) The maximum probability of measuring both atoms in the Rydberg state during a quench with average amplitudes 1 rad/$\mu$s (upper) and 3 rad/$\mu$s (lower) for each driving scenario. Markers indicate simulated results over 50$\mu$s, solid lines are calculated using the equations mentioned in the main text. Dotted lines show where the calculated values cross $P_{RR}=0.5$ and hence define the blockade radius $r_B$. (c) The simulated (markers) and calculated (solid lines) values of $r_B$ for a range of amplitudes. For the locally driven case the average amplitude is $\Omega$ and the ratio is $\Omega_2 / \Omega_1 = 3$
• Figure 2: Graph embeddings for unit disk graphs (top) and disk graphs (bottom) are evaluated using correlation matrices and maximum independence violation metrics. Both graphs have the same atom position, but local radii result in different connectivity. The radii, $r_{B, i}=0.98\sqrt[6]{C_6/\Omega_i}$, are implemented using a global drive of $\Omega_i=\pi$ for the unit disk graph (green open diamonds) or a local drive with $\Omega_i=\pi/20$ for $i\in (0, 4, 5)$ for the the disk graph (pink open triangles). A local drive implementation using the same set of local amplitudes but applied to random atoms (pink filled triangles) is shown as a control. A 100$\mu$s quench is performed with the atomic register scaled by $\lambda$, where $\lambda=\lambda_c$ (black dotted line) is the point at which the first edge in the graph is expected to break. Full disconnection of the unit disk graph also occurs at this point, whereas full disconnection of the disk graph is shown by the pink dotted line. The maximum correlation matrix is measured at connected ($\lambda/\lambda_c=0.8$) and disconnected ($\lambda/\lambda_c=2$) scaling values, and overlapped by red crosses which indicate edges in the graph.
• Figure 3: Relative sampling performance of local and global drives. Panel (a) displays the minimal non-unit-disk graph instances considered. For each instance, the parameters $(\kappa, \delta_f)$ are optimized to maximize $P_{\mathrm{MIS}}^{\mathrm{dr}}$. Panel (b) shows the relative sampling enhancement $\Delta_k = \frac{2\bigl(P_{\mathrm{MIS}-k}^{\,l} - P_{\mathrm{MIS}-k}^{\,g}\bigr)}{P_{\mathrm{MIS}-k}^{\,l} + P_{\mathrm{MIS}-k}^{\,g}},$ which compares the probabilities of preparing an independent set of size at least $\lvert\mathrm{MIS}\rvert - k$ under local and global driving. Annotated values indicate the corresponding violation probabilities $P_{\mathrm{violation}}^{g}$ and $P_{\mathrm{violation}}^{l}$, which quantify the probability of leaving the independent-set subspace. Note the difference in scale between the curves for $K_{1, 6}$ and those for the other graphs; this choice highlights the qualitative similarity of the trends while acknowledging their differing ranges. Across these non-unit-disk graphs, local driving systematically reduces $P_{\mathrm{violation}}$ and improves the sampling of near-optimal solutions up to $k \approx 2$. This illustrates that local addressability is important both for maintaining dynamics within the feasible independent-set subspace and for enhancing approximate MIS preparation.
• Figure 4: Blockade radius for systems with local simultaneous drive with different $\Omega_0$ and $\Omega_1$, obtained by simulation (circular points). Pulse durations for each simulation were chosen differently. Crosses mark data points where $\Omega_0=\Omega_1$ and solid lines indicate the value of $r_B^\text{L}$ found using Eq. \ref{['eq:rb_l']}.
• Figure 5: Gradient for systems with local simultaneous drive obtained by minimizing the distance from Eq. \ref{['eq:prr_pipulse']} to simulated data, shifted and rescaled according to $r \to (\Delta^{\text{L}}/\Delta^\pi)\cdot[r-r_B^{\text{L}}]+r_B^{\pi}$. Points of higher fluctuability are shown lighter.