  The polar decomposition of a matrix is a key element in the quantum linear
algebra toolbox. We show that the problem of quantum polar decomposition,
recently studied in Lloyd et al. [LBP+20], has a simple and concise
implementation via the quantum singular value transform (QSVT). We focus on the
applications to pretty-good measurements, a close-to-optimal measurement to
distinguish quantum states, and the quantum Procrustes problem, the task of
learning an optimal unitary mapping between given `input' and `output' quantum
states. By transforming the state-preparation unitaries into a block-encoding,
a pre-requisite for QSVT, we develop algorithms for these problems whose gate
complexity exhibits a polynomial advantage in the size and condition number of
the input compared to alternative approaches for the same problem settings
[LBP+20, GLMQW20]. For these applications of the polar decomposition, we also
obtain an exponential speedup in precision compared to [LBP+20], as the
block-encodings remove the need for the costly density matrix exponentiation
step. We contribute a rigorous analysis of the approach of [LBP+20].
