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Our systematic review explores quantum deep learning (QDL), an emerging deep learning sub-field, to assess whether quantum-based approaches outperform classical approaches in brain data learning tasks. This review is a pioneering effort to compare these deep learning domains. In addition, we survey neuroinformatics and its various subdomains to understand the current state of the field and where QDL stands relative to recent advancements. Our statistical analysis of tumor classification studies (n = 16) reveals that QDL models achieved a mean accuracy of 0.9701 (95% CI 0.9533\u20130.9868), slightly outperforming classical models with a mean accuracy of 0.9650 (95% CI 0.9475\u20130.9825). We observed similar trends across Alzheimer\u2019s diagnosis, stroke lesion detection, cognitive state monitoring, and brain age prediction, with QDL demonstrating better performance in metrics such as F1-score, dice coefficient, and RMSE. 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