A scalable quantum-enhanced greedy algorithm for maximum independent set problems

TL;DR

The paper tacklesMIS on regular graphs by marrying a shallow QAOA subroutine with a classical greedy strategy, using fixed-angle QAOA parameters derived from regular tree models to guide local decisions without parameter optimization. The approach yields a constructive, scalable, and hardware-friendly hybrid algorithm that scales linearly with graph size for fixed depth $p$, and shows practical performance gains over state-of-the-art classical MIS heuristics at depths as low as $p\approx 4$, evidenced by tensor-network simulations and experiments on a 20-qubit device. Tensor-network studies extend the analysis to graphs with up to $N=5000$, while hardware experiments on IQM Garnet validate that quantum guidance can improve greedy choices in the presence of noise. The work highlights a near-term pathway to quantum advantage: leverage quantum correlations for local decision-making within an efficient classical framework, with broad applicability to other local-structure optimization problems on sparse graphs.

Abstract

We investigate a hybrid quantum-classical algorithm for solving the Maximum Independent Set (MIS) problem on regular graphs, combining the Quantum Approximate Optimization Algorithm (QAOA) with a minimal degree classical greedy algorithm. The method leverages pre-computed QAOA angles, derived from depth-$p$ QAOA circuits on regular trees, to compute local expectation values and inform sequential greedy decisions that progressively build an independent set. This hybrid approach maintains shallow quantum circuit and avoids instance-specific parameter training, making it well-suited for implementation on current quantum hardware: we have implemented the algorithm on a 20 qubit IQM superconducting device to find independent sets in graphs with thousands of nodes. We perform tensor network simulations to evaluate the performance of the algorithm beyond the reach of current quantum hardware and compare to established classical heuristics. Our results show that even at low depth ($p=4$), the quantum-enhanced greedy method significantly outperforms purely classical greedy baselines as well as more sophisticated approximation algorithms. The modular structure of the algorithm and relatively low quantum resource requirements make it a compelling candidate for scalable, hybrid optimization in the NISQ era and beyond.

Figures (5)

• Figure 1: (A) First iteration of the hybrid quantum-classical greedy algorithm for QAOA depth $p=2$ on a 3-regular graph of $N=30$ nodes. The yellow node is selected as a node that has maximal $\ev{Z}_{p=2}$. The nodes of its $p=1$ subgraph are shown in green, the additional nodes in its $p=2$ subgraph in orange. The yellow node is added to the independent set and its direct neighbours (i.e. the nodes in the $p=1$ subgraph, the green nodes) are deleted from the graph. This deletion affects the expectation value of the remaining blue and orange nodes in the graph which need to be recomputed. (B) The $p=2$ subgraph that determines the expectation value. (C) The expectation values corresponding to the different subgraphs are computed using a superconducting 20 qubit IQM device.
• Figure 2: Performance of QAOA in solving the MIS problem on 3-regular graphs as a function of $1/p$ in the limit $N\rightarrow\infty$. We have performed two polynomial fits $a/p+b$ and $cp^d$ with corresponding fitting parameters $(a,b)=(-0.275,0.435)$ and $(c,d)=(-0.222,0.683)$ to indicate that these results suggest that QAOA needs significant depth to match the minimal greedy algorithm Frieze1994 (grey dashed line). We also compare to the performance of the linear-time algorithm of Ref. Marino2020 (SOTA, black dashed line) and an upper bound on the size of the independent set (UB, red dashed line) McKay1987Balogh2017.
• Figure 3: Independence ratios obtained by the quantum greedy algorithm compared with the independence ratios found by the greedy and linear-prioritized search algorithm for $d=3$ regular graphs. The dotted line corresponds to $r_{\infty}=0.445330$ which is the lower bound on the independence ratio in the limit $N\rightarrow \infty$ given in Ref. Marino2020. The error bars show three times the standard error of the mean originating from the average over graph instances. There are $200$ graph instances for QGreedy with each $p$, except for $p=3$ case with IQM Garnet, which has $100$ instances. The considered graph sizes are $N=50,100,200,500,1000,2000,5000$.
• Figure 4: Expectation values corresponding to all 75 possible subgraphs that can occur during the quantum-enhanced greedy search at depth $p=2$. The labels on x-axis denote the number of nodes and edges in each subgraph. Expectation values are displayed in order of descending number of nodes and ascending number of edges. The orange markers show the ideal results from tensor network simulations and blue markers show the corresponding results from the IQM Garnet quantum device. These expectation values are used to decide which node is added to the independent set: from all the subgraphs present in each iteration of the algorithm, the one with largest $\ev{Z_i}$ is chosen. The error bars mark standard deviation from $1000$ bootstrapped samples. They lie fully within the markers, signifying good statistical accuracy thanks to $20 000$ quantum circuit measurements for each data point.
• Figure 5: (A) Correlation plot between the noisy and ideal expectation values for $p=3$. The device measurements are shown in blue. The modeled values are shown in orange and are obtained via fitting Eq. \ref{['eq:error-model']} to the device measurements. The fit parameters are $\eta=0.03$, $\alpha=-0.05$ and $\sigma=0.04$. (B) The performance of the quantum-enhanced greedy algorithm under shrinking noise for $p=3$ and $N=1000$ as a function of $\eta$. The noise is modeled according to Eq. \ref{['eq:shrinking']}. (C) The performance of the quantum-enhanced greedy algorithm under realistic noise for $p=3$ and $N=1000$. The noise is modeled according to Eq. \ref{['eq:error-model']} with the same fit parameters as in (A). We consider 100 different random noise realizations of this model. We consider the same problem set of 200 graphs of size $N=1000$ in each of those. We have sorted the outcomes according to the achieved performances.