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Quantization Neural Networks (QNN) have attracted a lot of attention due to their high efficiency. To enhance the quantization accuracy, prior works mainly focus on designing advanced quantization algorithms but still fail to achieve satisfactory results under the extremely low-bit case. In this work, we take an architecture perspective to investigate the potential of high-performance QNN. Therefore, we propose to combine Network Architecture Search methods with quantization to enjoy the merits of the two sides. However, a naive combination inevitably faces unacceptable time consumption or unstable training problem. To alleviate these problems, we first propose the joint training of architecture and quantization with a shared step size to acquire a large number of quantized models. Then a bit-inheritance scheme is introduced to transfer the quantized models to the lower bit, which further reduces the time cost and meanwhile improves the quantization accuracy. Equipped with this overall framework, dubbed as Once Quantization-Aware Training~(OQAT), our searched model family, OQATNets, achieves a new state-of-the-art compared with various architectures under different bit-widths. In particular, OQAT-2bit-M achieves 61.6% ImageNet Top-1 accuracy, outperforming 2-bit counterpart MobileNetV3 by a large margin of 9% with 10% less computation cost. A series of quantization-friendly architectures are identified easily and extensive analysis can be made to summarize the interaction between quantization and neural architectures. Codes and models are released at https://github.com/LaVieEnRoseSMZ/OQA