With over 2.1 million new cases of breast cancer diagnosed annually, the incidence and mortality rate of this disease pose severe global health issues for women. Identifying the disease's influence is the only practical way to lessen it immediately. Numerous research works have developed automated methods using different medical imaging to identify BC. Still, the precision of each strategy differs based on the available resources, the issue's nature, and the dataset being used. We proposed a novel deep bottleneck convolutional neural network with a quantum optimization algorithm for breast cancer classification and diagnosis from mammogram images. Two novel deep architectures named three-residual blocks bottleneck and four-residual blocks bottle have been proposed with parallel and single paths. Bayesian Optimization (BO) has been employed to initialize hyperparameter values and train the architectures on the selected dataset. Deep features are extracted from the global average pool layer of both models. After that, a kernel-based canonical correlation analysis and entropy technique is proposed for the extracted deep features fusion. The fused feature set is further refined using an optimization technique named quantum generalized normal distribution optimization. The selected features are finally classified using several neural network classifiers, such as bi-layered and wide-neural networks. The experimental process was conducted on a publicly available mammogram imaging dataset named INbreast, and a maximum accuracy of 96.5% was obtained. Moreover, for the proposed method, the sensitivity rate is 96.45, the precision rate is 96.5, the F1 score value is 96.64, the MCC value is 92.97%, and the Kappa value is 92.97%, respectively. The proposed architectures are further utilized for the diagnosis process of infected regions. In addition, a detailed comparison has been conducted with a few recent techniques showing the proposed framework's higher accuracy and precision rate.