Closed-Loop Phase-Coherence Compensation for Superconducting Qubits Integrated Computational and Hardware Validation of the Aurora Method

We present an emulator-based and hardware feasibility study of Aurora-DD, a phase-coherence compensation method that integrates a sign-based feedback update of a global phase offset (Delta phi) with a fixed-depth XY8 dynamical decoupling (DD) scaffold. The feedback optimization is performed offline on a calibrated emulator and the resulting Delta phi* is deployed as pre-calibrated phase compensation on hardware. This represents an “offline closed-loop, online open-loop” feasibility demonstration. Using an Aer-based emulator calibrated with ibm_fez device parameters, Aurora-DD achieves substantial reductions in mean-squared error of the measured expectation value , yielding 68-97% improvement across phase settings phi = 0.05, 0.10, 0.15, 0.20 over n=30 randomized trials. These large-n emulator results provide statistically stable evidence that the combined effect of XY8 and Delta phi* suppresses both dephasing and systematic phase bias. On real superconducting hardware (ibm_fez), we perform a small-sample (n=3) multi-phase validation campaign. Aurora-DD yields point estimates corresponding to approximately 99.2-99.6% reduction in absolute error relative to a no-DD baseline across all tested phase points. These hardware numbers are reported transparently as feasibility evidence under tight queue and credit constraints. In contrast, the auxiliary Aurora+ZNE branch exhibits instability: shallow two-point ZNE occasionally amplifies calibration inconsistencies and produces large error outliers. We therefore relegate ZNE analysis to the Appendix and position Aurora-DD (without ZNE) as the primary contribution. Overall, the combined results support pre-calibrated Aurora-DD as a practical, stable, and hardware-compatible phase-coherence compensator for NISQ devices in single-qubit settings.

💡 Research Summary

This paper introduces Aurora‑DD, a hybrid error‑mitigation technique for single‑qubit superconducting circuits that combines a pre‑calibrated global phase offset (Δϕ*) with a fixed‑depth XY8 dynamical‑decoupling (DD) scaffold. The authors frame the method as an “offline closed‑loop, online open‑loop” experiment: the feedback optimization that determines Δϕ* is performed on a calibrated Aer emulator, and the resulting offset is then frozen and applied as a static correction on real hardware.

Theoretical foundation – Starting from the Bloch‑equation description of amplitude‑damping (T₁) and pure dephasing (T₂), the authors define a phase‑error proxy δZ(ϕ)=⟨Z⟩_ideal(ϕ)−⟨Z⟩meas(ϕ+Δϕ). The objective is to minimize the mean‑squared error J(Δϕ)=⟨δZ(ϕ)⟩². To solve this, a sign‑based gradient descent update Δϕ{k+1}=Δϕ_k+η·sgn(δZ_k) is employed, with η≤0.02 rad. This “sign‑SGD” rule is robust to stochastic measurement noise, maps naturally onto bang‑bang quantum control, and guarantees bounded rotations that avoid over‑rotation under realistic calibration drift (±6–8 % in T₂).

Emulator study – Using IBM fez calibration data (T₁≈155 µs, T₂≈110 µs, readout error maps), a noise model is built in Aer. For each of four rotation angles ϕ∈{0.05,0.10,0.15,0.20} rad, 30 independent random trials are executed for five configurations: baseline (no mitigation), DD‑only (XY8(12)), Δϕ‑only, Aurora‑DD (Δϕ*+XY8), and Aurora‑DD+ZNE (two‑point zero‑noise extrapolation). The optimizer converges to a single global offset Δϕ*≈0.15 rad that yields the largest average MSE reduction across all ϕ. Aurora‑DD achieves 68 %–97 % MSE reduction relative to baseline, clearly outperforming DD‑only and Δϕ‑only taken separately.

Hardware validation – Experiments are run on IBM’s superconducting backend ibm_fez (Heron‑r2 family). Each circuit uses one qubit, 2048 shots, and native gate compilation with pulse‑level control. Due to queue and credit constraints, only three repetitions per configuration are performed (n=3). Despite the small sample size, Aurora‑DD reduces absolute error by 99.2 %–99.6 % compared with the baseline, consistently beating both DD‑only and Δϕ‑only. The auxiliary Aurora‑DD+ZNE branch shows instability: shallow two‑point extrapolation sometimes amplifies calibration inconsistencies, producing large outliers. Consequently, ZNE is relegated to an appendix and not recommended for the current hardware regime.

Key insights

1. Global phase compensation works – The Bloch model predicts that low‑frequency dephasing manifests as an almost ϕ‑independent additive phase bias (ε_phase≪1). A single Δϕ* therefore corrects the dominant systematic error across a multi‑phase workload.
2. Closed‑loop offline optimization is practical – By performing the feedback loop on an emulator, the method avoids real‑time adaptive control, which is often infeasible on NISQ devices because of limited queue time, drift, and credit restrictions. The “offline closed‑loop, online open‑loop” paradigm yields a deterministic, low‑overhead control that can be deployed on any calibrated backend.
3. XY8 and phase compensation are synergistic – XY8 suppresses low‑frequency noise, while Δϕ* removes the residual static drift. Their combination yields error suppression beyond the sum of individual contributions.
4. ZNE is not universally beneficial – In the presence of deep DD sequences, even a simple two‑point extrapolation can destabilize results, highlighting the need for careful compatibility analysis when stacking mitigation techniques.

Limitations and future work – The study focuses on a single‑qubit setting; extending Aurora‑DD to multi‑qubit gates and correlated noise remains open. The global Δϕ* may over‑compensate when instantaneous drift deviates strongly from the emulator‑derived optimum; adaptive online refinement could address this. Larger hardware sample sizes would enable rigorous statistical inference, but the current work already demonstrates feasibility under realistic NISQ constraints.

Conclusion – Aurora‑DD, with a pre‑calibrated global phase offset and a modest XY8 scaffold, provides a stable, hardware‑compatible method to mitigate phase‑coherence errors on superconducting qubits. The offline closed‑loop optimization delivers a deterministic correction that can be applied without increasing circuit depth, making it attractive for near‑term quantum algorithms that demand high‑fidelity single‑qubit operations. The paper establishes a practical pathway for integrating simple feedback‑based control into existing quantum‑software stacks while cautioning against indiscriminate use of zero‑noise extrapolation in conjunction with dynamical decoupling.