Noise-adapted Quantum Error Correction for Non-Markovian Noise

TL;DR

The paper addresses quantum error correction under non-Markovian noise by leveraging the Petz recovery map, showing that the Petz map adapted to the full non-Markovian noise channel can safeguard the codespace and outperform standard stabilizer-based QEC, even at maximal noise. It generalizes AQEC conditions to Hermiticity-preserving non-CP maps and derives explicit fidelity bounds, illustrating oscillatory but non-vanishing worst-case fidelity due to information backflow. A detailed case study on non-Markovian amplitude damping demonstrates that NM Petz recovery yields superior fidelity compared to Markovian variants and stabilizer codes, while a Markovian Petz variant provides a practical alternative with slightly reduced fidelity and potential non-unitality. The work highlights practical implications for designing noise-adapted QEC in realistic, strongly coupled or structured environments and opens questions about the spectral properties of QEC superchannels and extensions to broader non-Markovian regimes.

Abstract

We consider the problem of quantum error correction (QEC) for non-Markovian noise. Using the well known Petz recovery map, we first show that conditions for approximate QEC can be easily generalized for the case of non-Markovian noise, in the strong coupling regime where the noise map becomes non-completely-positive at intermediate times. While certain approximate QEC schemes are ineffective against quantum non-Markovian noise, in the sense that the fidelity vanishes in finite time, the Petz map adapted to non-Markovian noise uniquely safeguards the code space even at the maximum noise limit. Focusing on the case of non-Markovian amplitude damping noise, we further show that the non-Markovian Petz map also outperforms the standard, stabilizer-based QEC code. Since implementing such a non-Markovian map poses practical challenges, we also construct a Markovian Petz map that achieves similar performance, with only a slight compromise on the fidelity.

Figures (6)

• Figure 1: Performance of different QEC schemes for non-Markovian AD noise with noise parameters $b=0.01$ and $\Gamma_0=5$ (see Appendix \ref{['sec:NMAD']}). We compare the bare qubit fidelity with that achieved by the $[[5,1,3]]$ code with stabilizer recovery and the $4$-qubit code with Petz recovery (Markovian and non-Markovian).
• Figure 2: Performance of different recovery schemes for the $[4,1]$ code subject to non-Markovian AD noise with the noise parameters $b=0.01$ and $\Gamma_0=5$ (see Appendix \ref{['sec:NMAD']}). For the Markovian Petz and Markovian Leung recovery, the recoveries are adapted to AD noise in the Markovian regime with $b=0.1$ and $\Gamma_0 =0.005$.
• Figure 3: Eigenvalues of the matrix $M$ in Eq.\ref{['eq:RE-matrixform']} (top graphs) and their time-derivatives ($\lambda'$) (bottom graphs) for Markovian Petz channel (adapted to AD with parameter $b=0.1$ and $\Gamma_0=0.005$) and non-Markovian Petz channel (adapted to AD with parameter $b=0.01$ and $\Gamma_0=5$) specific to the $4$-qubit code. The noise parameters are explained in Appendix \ref{['sec:NMAD']}.
• Figure 4: Eigenvalues of the $M$ matrix (top graph) and their derivatives (bottom graph), where the recovery is the stabilizer recovery for the $[[5,1,3]]$ code and the noise is the non-Markovian AD noise (see Appendix \ref{['sec:NMAD']}), with $b=0.01$ and $\Gamma_0=5$. This plot shows only the non-zero eigenvalues of the matrix $M$.
• Figure 5: Behavior of the worst-case fidelity under the non-unital combined channel $\mathcal{R} \circ \mathcal{E}$ for [[5,1,3]] (the Bold curve) and the [[4,1]] (the dotted curve) codes. Here, the Petz map $\mathcal{R}$ is adapted to an amplitude-damping process of strength $\gamma=0.1$, for both codes.

Theorems & Definitions (8)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof