Approaching Petz Recovery by Measurement-Free Quantum Signal Processing

The Petz recovery map is a near-optimal and widely used method in quantum information theory for approximating quantum error correction. However, its direct implementation is challenging, as it requires computing matrix square roots and inverses of density operators, as well as performing post-selection. In this work, we show that the Petz recovery map can be interpreted as a specific realization of a quantum search process, and can be implemented within the framework of generalized Grover search and quantum singular value transformation (QSVT) without post-selection. This provides a systematic way to asymptotically approach the Petz recovery map while trading off circuit complexity for accuracy. We further demonstrate an application of this method to scrambled quantum many-body dynamics, and compare the achieved fidelity with other recovery schemes.