Fast algorithm for quantum polar decomposition and applications

Abstract

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. [arXiv:2006.00841], 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 prerequisite 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 [Lloyd et al., arXiv:2006.00841; Gily\'en et al., arXiv:2006.16924]. For these applications of the polar decomposition, we also obtain an exponential speedup in precision compared to Lloyd et al. [arXiv:2006.00841], as the block-encodings remove the need for the costly density matrix exponentiation step. We contribute a rigorous analysis of the approach of Lloyd et al. [arXiv:2006.00841].