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The smart healthcare systems are transforming disease diagnosis by using robust computational technology, enabling timely, accurate, and customized clinical decision-making. Explainable Artificial Intelligence (XAI) has been introduced as a promising way to enhance the transparency, interpretability, and trustworthiness of artificial intelligence (AI) models, while Quantum Computing (QC) offers unmatched computational capacity to tackle complex healthcare optimization and machine learning (ML) problems. The convergence of XAI and QC can also considerably improve disease prediction through the integration of interpretable decision support and rapid data processing and optimization. This paper provides a short literature survey on the roles of XAI and QC in smart healthcare systems for disease diagnosis. It includes advanced AI methods like deep learning (DL) and machine learning (ML) models, along with top QC tools such as Qiskit, PennyLane, Cirq, and TensorFlow Quantum. It encompasses explainability techniques such as SHAP, LIME, and Grad-CAM, which are regarded as some of the most sophisticated in the world. The survey also discusses the use of these technologies in the prediction and diagnosis of diseases such as cancer, cardiovascular diseases, diabetes, neurological diseases, and infectious diseases. The paper provides a comparative examination of existing approaches, applications in the real world of healthcare, issues that are currently being faced, and future research priorities. The review suggests that quantum-assisted XAI is in its infancy but has immense potential to design intelligent, transparent, and efficient healthcare systems that can help promote reliable disease prediction and precision therapy. The insights provided in this study present a comprehensive understanding to the researchers and healthcare practitioners about emerging trends, research gaps, and feasible directions for future smart healthcare applications.