Petz Recovery through a Single Polar Transformation for Approximate Quantum Error Correction
Abstract
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.
Related
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BibTeX
@misc{zhao2026petz,
title = {{Petz Recovery through a Single Polar Transformation for Approximate Quantum Error Correction}},
author = {Hanzhang Zhao and Shenglong Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20515},
eprint = {2610.20515},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20515},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Petz Recovery through a Single Polar Transformation for Approximate Quantum Error Correction
AU - Hanzhang Zhao
AU - Shenglong Xu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20515
UR - https://arxiv.org/abs/2610.20515
AB - 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.
ER -