  Variational quantum algorithms (VQAs) that estimate values of widely used
physical quantities such as the rank, quantum entropies, the Bures fidelity and
the quantum Fisher information of mixed quantum states are developed. In
addition, variations of these VQAs are also adapted to perform other useful
functions such as quantum state learning and approximate fractional inverses.
The common theme shared by the proposed algorithms is that their cost functions
are all based on minimizing the quantum purity of a quantum state. Strategies
to mitigate or avoid the problem of exponentially vanishing cost function
gradients are also discussed.
