In the recent past, the success of Neural Architecture Search (NAS) has enabled researchers to broadly explore the design space using learning-based methods. Apart from finding better neural network architectures, the idea of automation has also inspired to improve their implementations on hardware. While some practices of hardware machine-learning automation have achieved remarkable performance, the traditional design concept is still followed: a network architecture is first structured with excellent test accuracy, and then compressed and optimized to fit into a target platform. Such a design flow will easily lead to inferior local-optimal solutions. To address this problem, we propose a new framework to jointly explore the space of neural architecture, hardware implementation, and quantization. Our objective is to find a quantized architecture with the highest accuracy that is implementable on given hardware specifications. We employ FPGAs to implement and test our designs with limited loop-up tables (LUTs) and required throughput. Compared to the separate design/searching methods, our framework has demonstrated much better performance under strict specifications and generated designs of higher accuracy by 18\% to 68\% in the task of classifying CIFAR10 images. With 30,000 LUTs, a light-weight design is found to achieve 82.98\% accuracy and 1293 images/second throughput, compared to which, under the same constraints, the traditional method even fails to find a valid solution.