Branch-Resolved Characterization of Feed-Forward Error in Dynamic Teleportation via Classical Choi Shadows

Abstract: Mid-circuit measurement and classical feed-forward are essential primitives for dynamic-circuit teleportation on superconducting quantum processors. However, the error associated with measurement-conditioned corrective operations remains poorly understood when evaluated with respect to individual measurement branches. In this paper, we present a framework for characterizing feed-forward error in dynamic circuit teleportation without losing valuable information related to its behavior across separate branches. We analyze three approaches to applying measurement-conditioned corrections: (i) physical application, (ii) post-processing adjustments, and (iii) a mitigated physical application which utilizes Bit-Flip Averaging (BFA)-based Probabilistic Readout Error Mitigation (PROM). We experimentally reconstruct branch Choi operators via an entangled reference qubit, and validate our physical-application and post-processing Choi-shadow estimators against full tomography of the branch Choi operators. We perform experiments on two physical qubit layouts which differ greatly in mid-circuit measurement readout error, and observe a reversal in the relative order in branch qualities obtained from the post-processing and PROM mitigation strategies. In one physical layout with higher measurement readout error, the operational feed-forward penalty is relatively modest (approximately 0.02-0.03) and PROM produces higher branch qualities than post-processing for every branch. In a separate layout with lower readout error, the operational feed-forward penalty increases to roughly 0.07, and post-processing exceeds PROM for all branch qualities. Our characterization framework can reveal branch-specific error structure and mitigation behavior that state-of-the-art outcome-averaged analyses fail to expose.

• The paper introduces a branch-resolved framework that characterizes feed-forward errors in dynamic quantum teleportation using classical Choi shadows.
• It evaluates three correction strategies—physical, post-processing, and PROM mitigation—by comparing estimator accuracy and branch quality under varying readout noise conditions.
• Numerical results on IBM's 156-qubit QPU confirm that PROM mitigation excels in high-noise regimes, whereas post-processing outperforms in low-error environments.

Branch-Resolved Characterization of Feed-Forward Error in Dynamic Teleportation

Motivation and Theoretical Framework

The paper "Branch-Resolved Characterization of Feed-Forward Error in Dynamic Teleportation via Classical Choi Shadows" (2604.28037) addresses the measurement-conditioned corrective operations (feed-forward) as an error source in dynamic quantum circuits. Employing quantum teleportation as a process model, the authors systematically dissect the channel dynamics resulting from mid-circuit measurements that induce branching—a quantum process that produces both classical and quantum outputs—leading to trace-non-increasing completely positive (CP-TNI) maps associated with each measurement outcome. The distinction from conventional outcome-averaged analyses is essential, as branch-resolved approaches can expose error structure masked in aggregate metrics.

The theoretical foundation is the quantum instrument framework, where each feed-forward branch is described by a branch map $E_m$. Choi-Jamiołkowski isomorphism underpins process characterization, and estimation of branch Choi operators from an entangled reference qubit enables robust, branch-resolved error metrics. Two types of W$_4$ resource states are deployed: "symmetric" and "perfect," differing in the degree of entanglement across the sender-receiver partition, thereby offering a controlled comparison for resource-dependent phenomena.

Correction Strategies and Experimental Implementation

Three feed-forward correction strategies are analyzed:

• Physical Application: Direct, branch-dependent Pauli corrections applied to Bob's qubit.
• Post-Processing Adjustments: Pauli corrections are tracked and applied classically, relabeling terminal outcomes according to the measurement record.
• PROM Mitigation: Bit-Flip Averaging (BFA) calibrates the mid-circuit readout error; Probabilistic Readout Error Mitigation (PROM) applies quasi-probability reweighting to correct for assignment errors via a symmetrized confusion matrix, targeting the ideal branch labels.

IBM's 156-qubit Fez superconducting QPU is the testbed. Two logical-to-physical qubit mappings select measurement pairs with high versus low readout assignment error, allowing systematic study of noise dependence. State preparation circuits for both W$_4$ resources are depth- and gate-matched, ensuring fair comparison.

Branch probabilities $p_m$, normalized branch Choi states $O_m$, branch qualities $q_m$, and statistical uncertainty via bootstrap resampling are comprehensively estimated. Choi-shadow estimators are validated against full two-qubit tomography, establishing quantitative fidelity and RMSE benchmarks.

Numerical Results

The Choi-shadow estimator aligns tightly with full tomography: for the perfect W$_4$ resource, RMSE decreases to 0.00694 and 0.00779 for physical and post-processing, respectively, in the primary observable family, confirming estimator reliability. For the symmetric W$_4$, corresponding values are slightly elevated but consistent.

Branch-resolved quality analysis reveals layout-dependent reversals in correction efficacy:

• High readout error (Layout 1): PROM outperforms post-processing across all branches; physical correction is worst. Feed-forward penalty is modest ($\sim$0.02–0.03).
• Low readout error (Layout 2): Post-processing surpasses PROM mitigation; physical correction remains lowest. Feed-forward penalty rises ($\sim$0.07), over 2.5$_4$0 the penalty in high-error layout.

PROM's mitigation effectiveness peaks in high-error layout (mean branch quality gain $_4$10.0307 for perfect W$_4$2) and diminishes in low-error layout ($_4$30.0175). This trend is consistent across both resource states.

Calibration data corroborate these trends: lower readout-error-free syndrome probability in noisy layout (0.9308) versus higher in quiet layout (0.9885).

Implications and Future Directions

The branch-resolved process characterization directly reveals structural error features inaccessible with outcome-averaged analyses. PROM mitigation is demonstrably advantageous in regimes with elevated readout noise but loses relative effectiveness as assignment error decreases, in which classical post-processing offers lower overhead and higher branch quality.

On practical grounds, this framework establishes the necessity of branch-resolved characterizations for evaluating dynamic quantum circuits, especially critical for error correction, qubit reuse, and adaptive operations reliant on measurement-conditioned feed-forward. Benchmarking correction and mitigation protocols in situ, with branch-resolved metrics, is key for deployment in scalable NISQ devices.

Theoretically, the methodology supports further exploration into dynamic-circuit error models and branch-specific noise tailoring. Extensions to more complex entanglement resources or multi-qubit teleportation (where full tomography is infeasible) will likely rely heavily on scalable classical shadow tomography.

For future AI-driven quantum error mitigation, branch-resolved data can inform reinforcement learning and automatic calibration schemes, offering channel-specific noise fingerprints. Progress in quantum process characterization will be contingent on integrating fine-grained, branch-resolved approaches such as those demonstrated in this work.

Conclusion

This paper develops a robust, branch-resolved framework for feed-forward error analysis in dynamic teleportation processes. Through Choi-shadow estimation and comparative correction strategies, it exposes channel-specific error and mitigation effects, demonstrating PROM's advantage in high noise and post-processing's superiority in low error environments. Branch-level characterization is shown to be essential for understanding and optimizing dynamic quantum circuits, providing tools and insights for both theoretical modeling and practical deployment in NISQ quantum processors (2604.28037).