The outliers in the clustering problem can easily affect the selection of cluster centers, and the expansion of the clustering scale will cause more computing resources to be consumed in the calculation of the distance between sample points. To address the above issues, a new quantum principal component analysis algorithm for clustering problems (QC-PCA) is proposed, improving the selection of the cluster center and the shortest distance search. In this paper, the principal components are marked by adding and subtracting thresholds to singular values and the cluster center is selected according to the potential function of the subset, thereby reduce the influence of abnormal points on the selection of the cluster center. In addition, a quantum minimum search algorithm is used to find the cluster center closest to the sample point, reducing the number of iterations required for clustering. Taking a small-scale data set as an example, the Cirq quantum programming framework is used to circuit design and simulation experiments. The experimental results show that compared with the existing quantum algorithms, the proposed QC-PCA algorithm improves the clustering accuracy. Performance analysis shows that compared with the existing classical and quantum algorithms, our algorithm has different degrees of improvement in the time complexity of the cluster center selection and the shortest distance search. And the resource consumption of the proposed QC-PCA algorithm is also lower than that of them.