Fast algorithm for quantum polar decomposition, pretty-good measurements, and the Procrustes problem

• URL: http://arxiv.org/abs/2106.07634v1
• Date: Mon, 14 Jun 2021 17:50:41 GMT
• Title: Fast algorithm for quantum polar decomposition, pretty-good measurements, and the Procrustes problem
• Authors: Yihui Quek and Patrick Rebentrost
• Abstract summary: We show that the problem of quantum polar decomposition has a simple and concise implementation via the quantum singular value QSVT. We focus on the applications to pretty-good measurements, a close-to-optimal measurement to distinguish quantum states, and the quantum Procrustes problem.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. [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].

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