Quantum algorithm for Petz recovery channels and pretty good measurements

TL;DR

This paper introduces a quantum algorithm for implementing Petz recovery channels and pretty good measurements, enabling near-optimal quantum state discrimination with improved efficiency using advanced quantum transformation techniques.

Contribution

It provides the first systematic quantum algorithm for Petz recovery channels and pretty good measurements, leveraging quantum singular value transformation and amplitude amplification.

Findings

Quantum algorithm efficiently implements Petz recovery channels.

Procedure for performing pretty good measurements with multiple state copies.

Algorithm's optimality is within a quadratic factor of the best possible.

Abstract

The Petz recovery channel plays an important role in quantum information science as an operation that approximately reverses the effect of a quantum channel. The pretty good measurement is a special case of the Petz recovery channel, and it allows for near-optimal state discrimination. A hurdle to the experimental realization of these vaunted theoretical tools is the lack of a systematic and efficient method to implement them. This paper sets out to rectify this lack: using the recently developed tools of quantum singular value transformation and oblivious amplitude amplification, we provide a quantum algorithm to implement the Petz recovery channel when given the ability to perform the channel that one wishes to reverse. Moreover, we prove that, in some sense, our quantum algorithm's usage of the channel implementation cannot be improved by more than a quadratic factor. Our quantum algorithm also provides a procedure to perform pretty good measurements when given multiple copies of the states that one is trying to distinguish.