Kernel method has a wide range of applications in machine learning. The combination of quantum computing and kernel method can effectively solve the problem of increasing computational costs in classical kernel method when the feature space becomes larger. Researches show that the minimized quantum circuits based on kernel method can be reliably executed on noisy intermediate-scale quantum devices. Some classifiers based on the quantum kernel method that have been proposed so far still have certain defects in terms of fully mapping data and circuit architecture. Therefore, we propose a compact quantum classifier based on polynomial kernel functions. First, a polynomial kernel function is introduced to increase the classification iteration rate of nonlinear data, thereby improve the classification efficiency. On this basis, a compact amplitude encoding is further proposed to encode the data labels corresponding to the quantum state. Compared with the existing quantum kernel method classifier, the number of coding bits of the quantum circuit of the proposed model can be reduced from 5 qubits to 3 qubits, and the two-qubit measurement in the existing method is simplified to a single-qubit measurement in the proposed model. In addition, the model achieves the optimal variance of the quantum circuit parameters in the measurement stage, which can effectively save computing resource overhead. Experimental simulations show that the expected value in the proposed classifier model is closer to the theoretical one, and higher classification accuracy is obtained. At the same time, the model has a low degree of entanglement, which effectively reduces the overhead of the entire preparation work.