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arXiv · 2006.00841

Quantum polar decomposition algorithm

Seth Lloyd·Samuel Bosch·Giacomo De Palma·Bobak Kiani·Zi-Wen Liu·Milad Marvian·Patrick Rebentrost·David M. Arvidsson-Shukur
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Abstract

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.

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Seth Lloyd, Samuel Bosch, Giacomo De Palma, Bobak Kiani, Zi-Wen Liu, Milad Marvian, Patrick Rebentrost, David M. Arvidsson-Shukur. 2020-06-01. Quantum polar decomposition algorithm. https://arxiv.org/abs/2006.00841

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