Deep neural networks are powerful and flexible models that have gained the attention of the machine learning community over the last decade. For a variety of tasks, they can even surpass human-level performance. Usually, to reach these excellent results, an expert spends significant time designing the neural architecture, with long trial and error sessions. In this scenario, there is a growing interest in automating this design process. To address the neural architecture search (NAS) problem, authors have presented new methods based on techniques such as reinforcement learning and evolutionary algorithms, but the high computational cost is still an issue for many of them. To reduce this cost, researchers have proposed to restrict the search space, with the help of expert knowledge. Quantum-inspired evolutionary algorithms present promising results regarding faster convergence. Motivated by this idea, we propose Q-NAS: a quantum-inspired algorithm to search for deep networks by assembling substructures. Q-NAS can also evolve some numerical hyperparameters, which is a first step in the direction of complete automation. We ran several experiments with the CIFAR-10 dataset to analyze the details of the algorithm. For many parameter settings, Q-NAS was able to achieve satisfactory results. Our best accuracies on the CIFAR-10 task were 93.85 percent for a residual network and 93.70 percent for a convolutional network, overcoming hand-designed models, and some NAS works. Considering the addition of a simple early-stopping mechanism, the evolution times for these runs were 67 GPU days and 48 GPU days, respectively. Also, we applied Q-NAS to CIFAR-100 without any parameter adjustment, reaching an accuracy of 74.23 percent, which is comparable to a ResNet with 164 layers. Finally, we present a case study with real datasets, where we used Q-NAS to solve the seismic classification task. In less than 8.5 GPU days, Q-NAS generated networks with 12 times fewer weights and higher accuracy than a model specially created for this task.