Universal syndrome-based recovery for noise-adapted quantum error correction

TL;DR

The paper tackles noise-adapted quantum error correction by addressing the core challenge that syndrome subspaces often overlap in approximate AQEC. It introduces a subspace orthogonalisation algorithm to produce an orthogonal set of error subspaces and a syndrome-based Petz recovery that leverages these subspaces, enabling syndrome measurements even when KL conditions are only approximately satisfied. Through analytical development and multiple code/nose examples (including Leung's [[4,1]] code, other four-qubit codes, and a six-qubit code), it demonstrates near-optimal entanglement fidelities and practical recovery strategies. Importantly, the authors implement a hardware-friendly version of the syndrome-based Petz map on IBM quantum processors, achieving substantial improvements in qubit lifetimes (T1) and providing a feasible path toward real-world, noise-adapted QEC on near-term devices.

Abstract

Quantum error correction (QEC) is an essential tool for quantum computing that enables reliable information processing in the presence of noise. Syndrome measurements play a central role in QEC, making it possible to unambiguously identify the location and type of errors. While syndrome extraction is natural for conventional QEC protocols, where the errors satisfy certain algebraic constraints \emph{perfectly}, this feature is largely missing in the framework of approximate or noise-adapted QEC. Rather, noise-adapted recovery maps like the Petz map are used in the latter scenario, but implementing such tailored recovery processes on the hardware can be quite challenging. Here, we address this issue by proposing an algorithmic approach to identifying error syndromes for arbitrary codes and noise processes. We then use our algorithm to develop a variant of the Petz recovery map -- a syndrome-based Petz recovery map -- which can then be implemented via syndrome measurements. We demonstrate the efficacy of our approach in the context of amplitude-damping noise, by constructing the syndrome-based Petz map for the $4$-qubit code. We execute our recovery circuits on IBM quantum hardware to successfully demonstrate break-even performance of a noise-adapted QEC protocol with upto a threefold improvement of the qubit $T_{1}$ times.

Figures (16)

• Figure 1: The figure shows the action of the noise channel on the codespace before and after orthogonalization. The ${A_k}$ denote the Kraus operators of the noise channel, while the ${E_k}$ represent the Kraus operators obtained through the orthogonalisation process. The relationship between ${E_k}$ and ${A_k}$ is given in Eq. \ref{['eq:Ek_forms']}.
• Figure 2: Schematics of the orthogonalisation algorithm, showing how the algorithm orthogonalises the overlapping subspaces.
• Figure 3: Comparison of the performance of various recovery protocols based on entanglement fidelity for the [[4,1]]-Leung code.
• Figure 4: Performances of different codes under the Petz and the syndrome-based Petz map. Here the [[4,1]]-optimal $\mathcal{R}_{P,\mathcal{E}}$ refers to the syndrome-based Petz recovery with the optimal [[4,1]]-code in Eq.\ref{['eq:liang_code']}. The $\mathcal{R}_{P,\mathcal{A}}$ is the ordinary Petz map, and the $R_{P,\mathcal{E}}$ is the syndrome-based Petz map. We also note the under the Petz recovery $\mathcal{R}_{P,\mathcal{A}}$ the entanglement fidelity is $F_{\rm ent} = 1- 1.5 \gamma^2$, which is better than the Leung code with $\mathcal{R}_{P,\mathcal{A}}$ (see the Table \ref{['tab:wcf_table']}).
• Figure 5: Performance of different recovery operation for the [[6,1,3]] code under the depolarizing noise process.

Theorems & Definitions (16)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof