Programmable switches empower stateful packet processing, in which incoming packets continuously update states in the data plane, while applications in the control plane read and write states. However, since the data plane and control plane are separated, a consistent view of states in both planes is required for stateful packet processing. Existing approaches suffer from either high latency or low accuracy. In this paper, we propose ApproSync, a framework that offers approximate state synchronization with low latency and high accuracy. To achieve low latency, ApproSync directly transfers states between switch ASICs and the control plane by bypassing switch operating systems. To achieve high accuracy, ApproSync utilizes the resources in the switch ASIC to realize rate control in state synchronization, such that it avoids potential state loss. It also bounds the divergence between the states in the data plane and that in the control plane under limited link capacity. We prototype ApproSync on Barefoot Tofino switches. The experimental results indicate that compared to existing approaches, ApproSync achieves order-of-magnitude latency reduction while maintaining high accuracy of state synchronization. Also, our experiments demonstrate that ApproSync provides significant latency benefits to existing network management applications and well preserves high application-level accuracy.