Towards Arbitrary QUBO Optimization: Analysis of Classical and Quantum-Activated Feedforward Neural Networks

TL;DR

This work tackles the challenge of solving arbitrary QUBO problems by introducing a memory-efficient, unsupervised FNN optimizer that delivers high-quality approximate solutions for large instances (e.g., 80-variable dense MaxCut and random QUBOs) in under a second on CPU and can outperform Gurobi on 200-variable random QUBOs within 100 seconds. It provides a detailed theory and hyperparameter framework for the FNN, and investigates a hybrid quantum-classical encoder-decoder (QCED) that uses a Rydberg annealer as a quantum activation. Results show that while QCED can offer faster convergence on some small problems, its quantum layer remains effectively inactive during training, leaving the classical FNN as the primary driver of performance; nonetheless, FNN demonstrates exceptional scalability and runtime advantages over traditional solvers for large QUBOs. The work suggests practical impact for real-time optimization tasks (e.g., smart grids, traffic, and finance) and outlines future directions, including Bayesian hyperparameter optimization and further exploration of quantum-classical hybrids.

Abstract

Quadratic Unconstrained Binary Optimization (QUBO) sits at the heart of many industries and academic fields such as logistics, supply chain, finance, pharmaceutical science, chemistry, IT, and energy sectors, among others. These problems typically involve optimizing a large number of binary variables, which makes finding exact solutions exponentially more difficult. Consequently, most QUBO problems are classified as NP-hard. To address this challenge, we developed a powerful feedforward neural network (FNN) optimizer for arbitrary QUBO problems. In this work, we demonstrate that the FNN optimizer can provide high-quality approximate solutions for large problems, including dense 80-variable weighted MaxCut and random QUBOs, achieving an average accuracy of over 99% in less than 1.1 seconds on an 8-core CPU. Additionally, the FNN optimizer outperformed the Gurobi optimizer by 72% on 200-variable random QUBO problems within a 100-second computation time limit, exhibiting strong potential for real-time optimization tasks. Building on this model, we explored the novel approach of integrating FNNs with a quantum annealer-based activation function to create a quantum-classical encoder-decoder (QCED) optimizer, aiming to further enhance the performance of FNNs in QUBO optimization.

Figures (10)

• Figure 1: (a) Exemplary FNN structure for solving 5-variable QUBO problems. (b) A schematic of QCED. The optimizer takes as input a QUBO matrix (Q) and consists of three major components: the encoder, the Rydberg annealer, and the decoder. In the final layer, the decoder returns a solution vector whose QUBO loss function is evaluated and minimized via gradient descent.
• Figure 2: (a) The optimization results measured in percentage errors for different powers $n$ used in the diagonal terms. The final output values are rounded to 0 or 1 to evaluate the energy cost. (b) The Shannon entropy of the return solution vector, which quantifies how close the continuous values are to either 0 or 1.
• Figure 3: Learning curves of FNN and QCED for 15-node MaxCut, 15-variable random QUBO, 15-node MWIS, and 4-city TSP (16 variables). The plots display the mean values for QCED and FNN with solid and dash-dotted lines, respectively. The one-standard-deviation margins are indicated by shaded areas.
• Figure 4: Optimization results of FNN and QCED in 500 iterations of training over 100 samples of MaxCut, random QUBO, MWIS, and TSP, each with varying numbers of variables. The error bars represent one standard deviation of the percentage error calculated across all the samples in the dataset.
• Figure 5: The average percentage change of the parameters in each layer of the encoder and decoder throughout the first 20 iterations of training. The magnitude of the change quantifies the activeness of each layer.