Syndrome-Based Petz Recovery Map

• The topic defines a syndrome-based Petz recovery map that uses orthogonalization to resolve overlapping error subspaces via iterative Gram–Schmidt procedures.
• The method constructs explicit recovery operators through syndrome extraction and correction, enabling noise-adapted protocols with reduced circuit depth.
• Experimental implementations on amplitude-damping noise models demonstrate significant improvements in logical qubit lifetimes and performance over traditional approaches.

A syndrome-based Petz recovery map is a quantum recovery operation that leverages syndrome measurement—orthogonal decomposition of error spaces—to implement a tailored version of the Petz map for noise-adapted quantum error correction. Unlike standard Petz recovery, which is agnostic to error syndromes, this approach constructs an explicit, syndrome-resolved recovery procedure for arbitrary codes and noise models, thereby enabling hardware-efficient, high-fidelity recovery protocols, especially in the regime of approximate error correction and noise-adapted coding (Biswas et al., 9 Oct 2025).

1. Algorithmic Orthogonalization of Error Syndromes

Conventional stabilizer code protocols exploit perfect Knill–Laflamme (KL) conditions, resulting in errors with orthogonal syndromes and enabling direct syndrome measurement. However, in noise-adapted or approximate codes (e.g., under amplitude-damping noise), Kraus operators generally do not partition the space into orthogonal syndrome subspaces. Implementing syndrome-resolved correction requires orthogonalizing the error subspaces generated by the channel Kraus operators $\{A_k\}$.

To achieve this, the paper introduces an iterative algorithm that transforms the original non-orthogonal Kraus operators $\{A_k\}$ into a new set $\{E_k\}$ satisfying

$P E_k^\dagger E_\ell P = 0 \qquad (k \neq \ell)$

with $P$ the projector onto the codespace. This is achieved by recursive Gram–Schmidt orthogonalization: $$E_1 P = A_1 P,\qquad E_k P = A_k P - \sum_{i=1}^{k-1} U_i P_i U_i^\dagger A_k P$$ where for each $i$, $U_i$ comes from the polar decomposition of $E_i P$ and $P_i$ is the projector onto the support of $P E_i^\dagger E_i P$. This procedure ensures that each resulting error $E_k$ projects the codespace into a mutually orthogonal syndrome subspace—satisfying a diagonalized, approximate Knill–Laflamme condition

$$\langle m_\ell| E_k^\dagger E_k | n_\ell \rangle = \alpha_k^{(mn)}\, \delta_{k\ell}.$$

This orthogonalization is crucial for syndrome-extractable recovery in non-stabilizer and approximate codes.

2. Construction of the Syndrome-Based Petz Recovery Map

With the set $\{E_k\}$, the syndrome-based Petz recovery map is defined as

$$\mathcal{R}_P^{(E)}(\cdot) = \sum_k P E_k^\dagger [E(P)]^{-1/2} (\cdot) [E(P)]^{-1/2} E_k P,$$

where $E(P) = \sum_k E_k P E_k^\dagger$ (restricted to the codespace support).

The explicit mutual orthogonality of the syndrome subspaces enables recovery by:

1. Measuring the syndrome (i.e., projecting onto the support of $E_k P$),
2. Correcting using the corresponding recovery operator $P E_k^\dagger [E(P)]^{-1/2}$,
3. Returning the state to the codespace.

This structure allows implementation as a hardware-efficient sequence: syndrome extraction followed by syndrome-targeted correction, in analogy with traditional stabilizer protocols but now adapted for general codes and noise.

3. Application to Amplitude-Damping Noise and Four-Qubit Leung Code

The paper applies the methodology to amplitude-damping (AD) noise $\mathcal{A}$ (Kraus operators $D_0$, $D_1$), with the four-qubit Leung code: $$|0_\ell\rangle = \frac{1}{\sqrt{2}} (|0000\rangle + |1111\rangle),\qquad |1_\ell\rangle = \frac{1}{\sqrt{2}} (|0011\rangle + |1100\rangle).$$ While the Leung code is AD-adapted, the action of physical damping errors (e.g., $D_0\otimes D_0\otimes D_0\otimes D_1$) results in overlapping error subspaces. The syndrome-orthogonalization algorithm produces a set of new errors $\{E_k\}$, enabling syndrome-resolved recovery. The table of recovery operators $(R_0, R_1, \dots, R_9)$ is constructed accordingly, and each operator is tied to a distinct, syndrome-measurable error sector.

Implementation proceeds by measuring primary stabilizer generators (e.g., $ZZII$, $IIZZ$) and secondary operators ($ZIII$, $IIIZ$), uniquely identifying the error syndrome, and applying the unitary correction $G_k$ appropriate for the detected syndrome. The circuits devised for these corrections act exclusively within the identified error subspaces, dramatically reducing overhead compared to full isometric or SDP-based implementations.

4. Circuit Depth, Hardware Efficiency, and Experimental Demonstration

The syndrome-based Petz map, being syndrome-resolved, leads to circuit structures with two key advantages:

• Circuit depth is substantially reduced relative to full Petz implementations that require large-scale unitary or Kraus extensions.
• Implementation exploits standard syndrome extraction tools: single-ancilla measurements, a lookup table for classical processing, and syndrome-specific correction unitaries.

The approach was benchmarked experimentally on IBM's quantum hardware (Heron and Eagle devices). For the amplitude-damping Leung code, the syndrome-based Petz map achieved a logical qubit $T_1$ lifetime that improved from $\sim337~\mu$s (bare) up to $\sim676~\mu$s (corrected), i.e., a twofold increase. In other device and cycle configurations, improvements by factors of three or more were observed.

5. Fidelity and Performance Analysis

Theoretical calculations for the four-qubit code under AD noise show:

• Syndrome Petz entanglement fidelity: $$F_{\text{ent}}(\mathcal{R}_P^{(E)}\circ\mathcal{A})=1-1.25\gamma^2+O(\gamma^3)$$,
• Worst-case fidelity: $1-1.15\gamma^2$,
• Demonstrated break-even performance, with lifetime improvements sometimes exceeding those of more general SDP-based recovery procedures, especially at moderate $\gamma$.

Experimental data corroborate these predictions, with tracking of logical fidelity vs. delay time showing significantly slower decay compared to both uncorrected bare qubits and unencoded equivalents.

6. Broader Implications and Universality

Syndrome-based Petz recovery, combined with noise-adapted code optimization, provides a universal and scalable template for implementing physically realistic, efficient, and robust recovery protocols in near-term quantum hardware:

• It generalizes both to numerical code search and to degenerate or non-stabilizer codes,
• The orthogonalization algorithm applies to arbitrary noise models, not limited to amplitude damping,
• Circuit design naturally adapts to syndrome measurement hardware and feedback capabilities,
• Enables principled reduction of circuit complexity and depth, making it feasible for noisy intermediate-scale quantum (NISQ) processors with limited coherence and gate fidelity.

This framework bridges foundational theory and implementation, and is likely to inform practical fault-tolerant quantum computing architectures in the noise-adapted regime (Biswas et al., 9 Oct 2025).

---

Table: Summary of Syndrome-Based Petz Map Construction

Step	Methodological Element	Purpose
Orthogonalization	Gram–Schmidt or polar decomposition	Extract orthogonal syndrome subspaces
Syndrome measurement	Projective measurements	Identify the error syndrome in hardware
Recovery operation	Syndrome-dependent Petz correction	Apply targeted unitary to codespace
Hardware realization	Lookup-table and feedback circuits	Resource-efficient near-term implementation
Performance evaluation	Worst-case and entanglement fidelity	Quantify recovery and logical lifetime

---

This approach marks a shift from abstract, unstructured Petz recovery to physically realizable, universal syndrome-based protocols for quantum error correction, merging rigorous quantum information theory with device-level applicability.