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arXiv:2108.10893 · Communications in Mathematical Physics

Approximate Petz Recovery from the Geometry of Density Operators

Sam Cree, Jonathan Sorce

Published Mar 18, 2022• 5 citations
arXivHTMLPDF
DOI
10.1007/s00220-022-04357-2

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The Petz recovery map provides a near optimal decoder for approximate quantum error correction, but its inverse square-root normalization complicates circuit implementation. We use the polar structure of the Petz recovery isometry to implement recovery through a single quantum singular value transformation (QSVT). We give explicit block-encoding circuits and an extended recovery isometry that yields approximate Petz recovery without postselection. We bound the diamond distance between the approximate and ideal Petz encoding-recovery channels in terms of a spectrum-weighted loss, which provides a channel specific objective for optimizing the QSVT angles. Uniform polynomial approximation yields an asymptotic error bound that decays exponentially with the number of QSVT iterations, at a rate controlled by the smallest nonzero scaled singular value. Applications to Haar-random encoding and the Hayden-Preskill protocol illustrate the tradeoff between circuit depth and recovery accuracy and place existing scrambling decoders within a common Petz recovery framework.
Optical systems are a main platform for quantum information processing. A main challenge is information loss due to scattering in unmonitored modes. These losses are modeled as state-independent beam-splitter interactions, with a thermal state (for all practical purposes, the vacuum) in the second input port. The perfect correction of these Gaussian lossy channels with Gaussian operations alone is known to be impossible. In this work, we investigate the Petz recovery map as an approximate recovery. For single mode losses and Gaussian reference states, the Petz map is found to use either a beam-splitter or a state-independent amplifier, depending on the parameters. Then we study the recovery performance on several examples, showing that it is near-optimal among the considered class of protocols. We also obtain more specific comparisons: Petz is always better than just re-preparing the reference state; but it is worse than doing nothing if the reference state is far from the true state. Finally, we extend our study to losses on two modes, and compare the global Petz map to the local implementation on each mode separately.
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 Letter 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.