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Cancer classification based on gene expression increases early diagnosis and recovery, but high-dimensional genes with a small number of samples are a major challenge. This work introduces a new hybrid quantum kernel support vector machine (QKSVM) combined with a Binary Harris hawk optimization (BHHO) based gene selection for cancer classification on a quantum simulator. This study aims to improve the microarray cancer prediction performance with the quantum kernel estimation based on the informative genes by BHHO. The feature selection is a critical step in large-dimensional features, and BHHO is used to select important features. The BHHO mimics the behavior of the cooperative action of Harris hawks in nature. The principal component analysis (PCA) is applied to reduce the selected genes to match the qubit numbers. After which, the quantum computer is used to estimate the kernel with the training data of the reduced genes and generate the quantum kernel matrix. Moreover, the classical computer is used to draw the support vectors based on the quantum kernel matrix. Also, the prediction stage is performed with the classical device. Finally, the proposed approach is applied to colon and breast microarray datasets and evaluated with all genes and the selected genes by BHHO. The proposed approach is found to enhance the overall performance with two datasets. Also, the proposed approach is evaluated with different quantum feature maps (kernels) and classical kernel (RBF).