QAOA-Predictor: Forecasting Success Probabilities and Minimal Depths for Efficient Fixed-Parameter Optimization

TL;DR

This work proposes a novel approach using a Graph Neural Network (GNN) to predict QAOA performance: Based on a graph representation of the problem, the GNN forecasts the probability of the optimal solution in the resulting distribution across different parameter initializations and layer depths for a wide variety of combinatorial optimization problems.

Abstract

Quantum Computing promises to solve complex combinatorial optimization problems more efficiently than classical methods, with the Quantum Approximate Optimization Algorithm (QAOA) being a leading candidate. Recent fixed-parameter variations of QAOA eliminate costly run-time optimization, but determining their optimal initialization as well as the number of required layers (p) for a target solution remains a critical, unsolved challenge. In this work, we propose a novel approach using a Graph Neural Network (GNN) to predict QAOA performance: Based on a graph representation of the problem, the GNN forecasts the probability of the optimal solution in the resulting distribution across different parameter initializations and layer depths for a wide variety of combinatorial optimization problems. We demonstrate that the GNN accurately predicts QAOA performance within a 10% margin of the true values. Furthermore, the model exhibits strong generalization capabilities across unseen problem classes, larger problem sizes, and higher layer counts. Our approach allows to identify viable problem instances for QAOA and to select an adequate parameter initialization strategy with minimal layer depth, without the need of costly parameter optimization.

Figures (15)

• Figure 1: Overview of our method: For a specific problem, its QUBO matrix is generated, which is subsequently transformed into a (normalized) Ising Hamiltonian (i). The Hamiltonian matrix is then re-formulated as a graph (ii), which is fed to a Graph Neural Network (GNN) (iii) that outputs a graph-embedding (iv). Then, a Multi-Layer Perceptron (MLP) receives this graph-embedding together with values of $\Delta_{\gamma,\beta}$ and the numbers of layers $p$, and predicts the performance of LR-QAOA for this problem instance (v), measured as the probability of the optimal solution in the resulting probability distribution. This prediction can than be used to initialize LR-QAOA with the respective parameters (vi) and sample the optimal solution with the predicted probability (vii) without the need of additional parameter optimization.
• Figure 2: Normalized QUBO matrices of the combinatorial optimization problems considered in this work.
• Figure 3: Performance of LR-QAOA with fixed $\Delta_{\gamma, \beta}=0.3$ parameter on two exemplary problem classes. For each problem class (Max-Cut and Knapsack) and size $N \in [5, 19]$, 150 random instances are sampled and the mean probability of the optimal solution across these samples is plotted for increasing number of layers $p$. For the Max-Cut problem class (left), a problem size of $N=7$ requires $p=20$ layers for a $80\%$ probability of sampling the optimal solution, while a larger instance of $N=14$ requires $p=60$ layers to achieve the same probability. For the Knapsack problem class (right), the effect of compensating size via layers is not as strongly pronounced. Here the overall probability is not only lower, but also more layers are required to increase the probability of the optimal solutions.
• Figure 4: The GNN and CNN models share a similar overall structure, differing only in their input format and initial processing layers. The GNN uses a graph-based backbone to process problem instances as graphs, while the CNN uses a convolutional backbone to process them as matrices.After this initial feature extraction, both models use the same regression head. This head concatenates the backbone's embedding with the parameter $\Delta_{\gamma,\beta}$, and the layer number $p$. This combined vector is then passed through a MLP to predict the final success probability.
• Figure 5: Embedding of the validation data set using PCA colored by class (left) and size (right)