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

TL;DR

This paper introduces a quantum algorithm for polar decomposition using quantum singular value transform, enabling efficient solutions for quantum state discrimination and unitary learning with polynomial and exponential speedups, respectively.

Contribution

It provides a concise quantum polar decomposition method via QSVT, improving efficiency and precision for quantum measurements and unitary approximations over previous approaches.

Findings

Polynomial advantage in size and condition number

Exponential speedup in precision

Simplified implementation of quantum polar decomposition

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].