Solving Larger Maximum Clique Problems Using Parallel Quantum Annealing

TL;DR

The paper tackles solving large Maximum Clique problems on hardware-limited quantum annealers by coupling exact classical graph decomposition (DBK) with parallel quantum annealing (tiling) on D-Wave’s Advantage system. It shows that decomposing graphs into subproblems and solving them in parallel enables exact or near-optimal results for graphs up to 120 vertices, with strong performance on dense graphs at small subproblem cutoffs. Key contributions include the DBK-pQA hybrid architecture, a formalized TTS framework for parallel subproblems, and experimental evidence that, in certain regimes, the quantum-assisted approach can vastly outperform a state-of-the-art classical solver FMC. The work demonstrates practical pathways to leverage parallel quantum annealing for larger combinatorial problems and lays groundwork for extending hybrid quantum-classical strategies to other NP-hard graph problems.

Abstract

Quantum annealing has the potential to find low energy solutions of NP-hard problems that can be expressed as quadratic unconstrained binary optimization problems. However, the hardware of the quantum annealer manufactured by D-Wave Systems, which we consider in this work, is sparsely connected and moderately sized (on the order of thousands of qubits), thus necessitating a minor-embedding of a logical problem onto the physical qubit hardware. The combination of relatively small hardware sizes and the necessity of a minor-embedding can mean that solving large optimization problems is not possible on current quantum annealers. In this research, we show that a hybrid approach combining parallel quantum annealing with graph decomposition allows one to solve larger optimization problem accurately. We apply the approach on the Maximum Clique problem on graphs with up to 120 nodes and 6395 edges.

Figures (8)

• Figure 1: Illustration of the DBK vertex splitting applied to vertex $v$, resulting in the induced subgraph $G_v$ of $v$, and a graph $G'=(V',E')$ with $V'=V \setminus \{v\}$ and all edges incident to $v$ removed from $E$. Figure taken from Pelofske2019mc.
• Figure 2: Disjoint minor embeddings for parallel problem solving on the quantum annealer. The chip topology is for the D-Wave Advantage System 4.1. The minor-embeddings are of cliques of sizes $N \in \{100, 90, 80, 70, 60, 50\}$ from top left to bottom. Red and Blue coloring is used (randomly) in order to help to visually differentiate neighboring minor embeddings.
• Figure 3: Number of subgraphs (top left), average subgraph density (top right), and average subgraph size (bottom) at each cutoff level of the DBK-fmc algorithm. Log scale on the y-axis of the top left plot. Input graph densities ranging from $0.1$ to $0.9$ (see color legend).
• Figure 4: Scatter plot of the Maximum Clique approximation ratios across all subgraphs that were solved as a function of the DBK cutoff. Computed by taking the best Maximum Clique solution (unembedded with majority vote) out of the $1,000$ samples used for each subgraph. The data at a DBK cutoff of $120$ corresponds to the original input graphs.
• Figure 5: Left: Failure rate (failure to reach a ground state solution) as a function of the DBK cutoff values (after majority vote unembedding). Graph densities ranging from $0.1$ to $0.9$ (see color legend). Right: Averaged results for four groups of input graph densities ; $0.1-0.3$, $0.3-0.5$, $0.5-0.7$, and $0.7-0.9$ (the coloring corresponds to the median graph density of those ranges).