Quantum Optimization on Rydberg Atom Arrays with Arbitrary Connectivity: Gadgets Limitations and a Heuristic Approach

TL;DR

This work analyzes the complexity-theoretic limits of polynomial reductions from arbitrary graphs to unit-disk instances and proposes a divide-and-conquer heuristic with only linear overhead which leverages precalibrated atomic layouts.

Abstract

Programmable quantum systems based on Rydberg atom arrays have recently emerged as a promising testbed for combinatorial optimization. Indeed, the Maximum Weighted Independent Set problem on unit-disk graphs can be efficiently mapped to such systems due to their geometric constraints. However, extending this capability to arbitrary graph instances typically necessitates the use of reduction gadgets, which introduce additional experimental overhead and complexity. Here, we analyze the complexity-theoretic limits of polynomial reductions from arbitrary graphs to unit-disk instances. We prove any such reduction incurs a quadratic blow-up in vertex count and degrades solution approximation guarantees. As a practical alternative, we propose a divide-and-conquer heuristic with only linear overhead which leverages precalibrated atomic layouts. We benchmark it on Erdös-Rényi graphs, and demonstrate feasibility on the Orion Alpha processor.

Figures (5)

• Figure 1: Summary of the paper. We solve the Maximum Weighted Independent Set (MWIS) problem on Rydberg Atoms arrays. $(a)$ The native graphs that can be embedded in the machine are Unit-Disk (UD) graphs, for which a Polynomial-Time Approximation Scheme (PTAS) exists. MWIS problems on UD graphs exemplify an 'easy subset' of NP-hard problems to solve for classical algorithms. Generic not embeddable graphs exhibit long-range couplings that violate the UD constraints. In the example, the edges that do not comply with the UD constraint are highlighted in red. To tackle the MWIS on arbitrary graphs one must employ reductions that allow to embed it on native layouts. Other than incurring in a quadratic overhead in mapping from the original graph $G$ to $G'$, such reductions do not preserve the solution quality. $(b)$ Solutions in $G'$ with gaps $\Delta(G')$ larger than a threshold $\bar{\Delta}(G')$ (dashed vertical line) map to poor quality solutions of the problem in $G$ -- comparable to random sampling. $\bar{\Delta}(G')$ goes to zero $\propto 1/n$. $(c)$ We propose a heuristic algorithm for solving the MWIS problem. This approach splits arbitrary graphs into smaller instances of the MWIS on precalibrated atomic layouts, it guarantees feasible solutions and incurs an $\mathcal{O}(n)$ overhead. We plot the gap to the best solution computed by means of CPLEX cplex2024 and Simulated annealing (SA) dimod We show results of both quantum simulations (Q-sim, in orange) and experiments (QPU, in green) by employing our method, in comparison with the solution obtained by a Greedy algorithm (purple). We consider Erdös-Rényi graphs of size up to $n=500$ and $p=0.5$ (probability of drawing an edge between two nodes). The simulation results are averaged over 100 instances while the experimental results over 10. The error bars correspond to the standard deviation of the mean. In the inset of $(c)$, we show an histogram counting the occurrences of the largest subgraph obtained with our method as a function of its size $n'$.
• Figure 2: Annealing protocol of time dependent controls $\Omega$ and $\delta$ employed to find the MIS of the graphs. The values and the positions of the interpolated points are written on the figure. We report them here for convenience: $(t,\Omega,\delta)\in$$\{(0,0,-15.50\pi)$, $(1000,1.54\pi,-7.56\pi)$, $(2000,4\pi,0.77\pi)$, $(3000,3.78\pi,4.25\pi)$, $(4000,0,9.89\pi)\}$
• Figure 3: Single iteration of the Greedy Lattice Subgraph mapping. We pick a random node $n_0$ in the original graph $G$ and $l_0$ in the lattice $L$. We grow the mapping exploring the neighborhood of the selected node, till no more assignments are possible.
• Figure 4: Here we plot on which geometry the 10% largest subgraphs created during the GLS mapping starting from for Erdös-Rényi graphs would be mapped. We observe that the largest subgraphs are more frequently mapped on Triangular lattices.
• Figure 5: We plot the size of the MIS obtained via quantum simulations (Q-sim) by employing our method in comparison with the solution obtained by a Greedy algorithm, CPLEX cplex2024 and Simulated annealing (SA), for Erdös-Rényi graphs of size up to $n=500$. We consider the probability of drawing an edge between two nodes as $(a)$$p=0.25$; $(b)$$p=0.75$. The results are averaged over 100 instances. The error bars correspond to the standard deviation of the mean.