Embedding the MIS problem for non-local graphs with bounded degree using 3D arrays of atoms

TL;DR

This paper introduces a deterministic polynomial-time method to embed non-local graphs with bounded degree into 3D arrays of atoms, advancing quantum approaches to solving complex combinatorial problems like MIS beyond local graph classes.

Contribution

The authors develop a novel deterministic embedding technique for non-local bounded-degree graphs into 3D atomic arrays, enabling quantum algorithms to address a broader class of problems.

Findings

Embedding method is polynomial-time and deterministic.

Enables encoding of non-local graphs in atomic arrays.

Facilitates quantum solutions for complex combinatorial problems.

Abstract

In the past years, many quantum algorithms have been proposed to tackle hard combinatorial problems. These algorithms, which have been studied in depth in complexity theory, are at the heart of many industrial applications. In particular, the Maximum Independent Set (MIS) is a known NP-hard problem that can be naturally encoded in Rydberg atom arrays. By representing a graph with an ensemble of neutral atoms one can leverage Rydberg dynamics to naturally encode the constraints and the solution to MIS. However, the classes of graphs that can be directly mapped node-to-atom on such devices are limited to Unit-Disk graphs. In this setting, the inherent locality of the graphs can be leveraged by classical polynomial-time approximation schemes (PTAS) that guarantee an {\epsilon}-approximate solution. In this work, we present a deterministic and polynomial-time construction to embed a large family of non-local graphs in 3D atomic arrays. This construction is a first crucial step towards tackling combinatorial tasks on quantum computers for which no classical efficient {\epsilon}-approximation scheme exists.