Approximate combinatorial optimization with Rydberg atoms: the barrier of interpretability

TL;DR

The paper analyzes interpretability barriers when solving MIS via Rydberg-atom embeddings, focusing on Crossing Lattice embeddings that map MIS to UD-MWIS and examining two post-processing strategies: a computationally intense distance-to-nearest-interpretable-state method and a cheap deselection method. It shows that non-interpretable states proliferate above a domain-wall energy threshold and that defect density dictates asymptotic performance, leading to an amplification of approximation errors and challenging prospects for scalable quantum advantage under PCP-type hardness. The work derives lower bounds for the deselection approach and universal DoS patterns that persist across embeddings, arguing that embedding-based PTAS-like improvements are unlikely and prompting a shift toward heuristics. Overall, the results highlight fundamental interpretability costs in current Rydberg-embedding schemes and point to exploring alternative algorithms or heuristic strategies for UD-MWIS problems.

Abstract

Analog quantum computing with Rydberg atoms is seen as an avenue to solve hard graph optimization problems, because they naturally encode the Maximum Independent Set (MIS) problem on Unit-Disk (UD) graphs, a problem that admits rather efficient approximation schemes on classical computers. Going beyond UD-MIS to address generic graphs requires embedding schemes, typically with chains of ancilla atoms, and an interpretation algorithm to map results back to the original problem. However, interpreting approximate solutions obtained with realistic quantum computers proves to be a difficult problem. As a case study, we evaluate the ability of two interpretation strategies to correct errors in the recently introduced Crossing Lattice embedding. We find that one strategy, based on finding the closest embedding solution, leads to very high qualities, albeit at an exponential cost. The second strategy, based on ignoring defective regions of the embedding graph, is polynomial in the graph size, but it leads to a degradation of the solution quality which is prohibitive under realistic assumptions on the defect generation. Moreover, more favorable defect scalings lead to a contradiction with well-known approximability conjectures. Therefore, it is unlikely that a scalable and generic improvement in solution quality can be achieved with Rydberg platforms -- thus moving the focus to heuristic algorithms.

Figures (19)

• Figure 1: General procedure to solve MIS on a generic graph by the means of an adiabatic evolution on a UD graph. Red: selected vertices after the UD-MWIS exact resolution.
• Figure 2: Structure of the initial and embedding solution spaces for a given embedding scheme, with sub-spaces of solutions characterized by their Hamming distance $d = 0, 1, \dots$ to the nearest interpretable solution.
• Figure 3: Path embedding of a single-vertex graph into two interpretable configurations (green highlight), with selected vertices in red. Due to domain walls (red highlight), some configurations with a low energy have a high Hamming distance $d$ to the closest interpretable configuration.
• Figure 4: (Top) Low-energy DoS in the path graph $P_{40}$ with $1/w \in \{4, 20\}$. On the left, the two interpretable states have a gray outline to enhance readability. (Bottom) Distribution of $\tau_d$ in increasing energy windows, with an exponential decrease of $\tau_0(E)$ (dotted line).
• Figure 5: (a) Initial graph with $N = 4$ vertices, $\vert E\vert = 3$ edges (plain lines) and $N(N-1)/2 - \vert E \vert = 3$ pairs of non-adjacent vertices (dashed lines). (b) Corresponding CL embedding, with $N$ even-length paths used to embed the $N$ initial vertices and interconnected by $N(N-1)/2$ gadgets. Colors identify the mapping between initial vertices and paths in the CL. (c) Legend, with two types of gadgets subgraphs representing initial edges (plain diamond) or their absence (dotted diamond).