Cancer classification based on gene expression increases early diagnosis and recovery, but high-dimensional genes with a small number of samples are considered the major challenge in cancer classification tasks. Gene selection is a critical step in large-dimensional features to select the most informative and relevant subset of genes. The Gene selection problem utilizes metaheuristic techniques to extract the optimal subset of genes in microarray cancer datasets, such as particle swarm optimization (PSO), Harris hawks optimization (HHO), and grey wolf optimizer (GWO). Binary Harris hawk optimization (BHHO) technique, which mimics the behavior of the cooperative action of Harris hawks in nature, is recently proposed as one of these techniques. The biggest challenge of quantum hardware is the limited number of qubits, which restricts the use of quantum devices in real applications. The principal component analysis (PCA) is applied to reduce the selected genes to match the qubit numbers. In this work, a new hybrid quantum-kernel support vector machine (QKSVM) with BHHO called BHHO-PCA-QKSVM 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 quantum computer is used for estimation the kernel with the training data of the reduced genes and generation of the quantum kernel matrix. Besides, the classical computer is used for drawing the support vectors based on the quantum kernel matrix and applying the prediction stage. The colon and breast microarray datasets are used for evaluating the proposed approach performance with all genes and the selected genes. The proposed model enhances the overall performance of the two datasets. Also, the final results of the proposed model compared with different quantum feature maps (kernels) and classical radial basis function (RBF) kernel.