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 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.

中文翻译：

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Petz 恢复通道的量子算法和非常好的测量

Petz 恢复通道作为一种近似逆转量子通道效应的操作，在量子信息科学中发挥着重要作用。相当好的测量是 Petz 恢复通道的一个特例，它允许近乎最佳的状态鉴别。这些吹嘘的理论工具的实验实现的障碍是缺乏系统和有效的方法来实现它们。本文着手弥补这一不足：使用最近开发的量子奇异值变换和不经意幅度放大工具，我们提供了一种量子算法来实现 Petz 恢复通道，当能够执行人们希望反转的通道时。此外，我们证明了我们的量子算法' 信道实现的使用不能超过二次因子的改进。我们的量子算法还提供了一个程序，当给定一个人试图区分的多个状态副本时，可以执行非常好的测量。