Quantum linear algebra methods, in particular block-encoding and quantum singular value transformation, provide efficient tools for processing coherent quantum information natively. Our new understanding and new tools enable us to solve several related problems in greater generality or more efficiently than what was known before. I will illustrate the applicability of these tools for different problems including estimation, testing and tomography tasks related to quantum states as well as for the implementation of Petz recovery channels -- a quantum generalization of Bayes' inference.
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