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The transformer architectures with attention mechanisms have obtained success in Nature Language Processing (NLP), and Vision Transformers (ViTs) have recently extended the application domains to various vision tasks. While achieving high performance, ViTs suffer from large model size and high computation complexity that hinders the deployment of them on edge devices. To achieve high throughput on hardware and preserve the model accuracy simultaneously, we propose VAQF, a framework that builds inference accelerators on FPGA platforms for quantized ViTs with binary weights and low-precision activations. Given the model structure and the desired frame rate, VAQF will automatically output the required quantization precision for activations as well as the optimized parameter settings of the accelerator that fulfill the hardware requirements. The implementations are developed with Vivado High-Level Synthesis (HLS) on the Xilinx ZCU102 FPGA board, and the evaluation results with the DeiT-base model indicate that a frame rate requirement of 24 frames per second (FPS) is satisfied with 8-bit activation quantization, and a target of 30 FPS is met with 6-bit activation quantization. To the best of our knowledge, this is the first time quantization has been incorporated into ViT acceleration on FPGAs with the help of a fully automatic framework to guide the quantization strategy on the software side and the accelerator implementations on the hardware side given the target frame rate. Very small compilation time cost is incurred compared with quantization training, and the generated accelerators show the capability of achieving real-time execution for state-of-the-art ViT models on FPGAs.