Quantum polar decomposition algorithm
Seth Lloyd, Samuel Bosch, Giacomo De Palma, Bobak T. Kiani, Zi-Wen Liu, Milad Marvian, Patrick Rebentrost, David R. M. Arvidsson-Shukur · arXiv (Cornell University) · 2020
The polar decomposition for a matrix $A$ is $A=UB$, where $B$ is a positive Hermitian matrix and $U$ is unitary (or, if $A$ is not square, an isometry). This paper shows that the ability to apply a Hamiltonian $\pmatrix{ 0 & A^\dagger \cr A & 0 \cr} $ translates into the ability to perform the transformations $e^{-iBt}$ and $U$ in a deterministic fashion. We show how to use the quantum polar decomposition algorithm to solve the quantum Procrustes problem, to perform pretty good measurements, to find the positive Hamiltonian closest to any Hamiltonian, and to perform a Hamiltonian version of the quantum singular value transformation.