We derive analytically that the optimal feedforward phase for ZZ-coupling correction in superconducting qubits is given by the argument of the first circular moment of the accumulated phase distribution: φ* = arctan(S/C), where C = ⟨cos φ⟩ and S = ⟨sin φ⟩. A negative second derivative proves this is always a maximum. We measure GPS = P_ff − P_ref on IBM Heron r2 processors (ibm_kingston, ibm_marrakesh; five qubit pairs) and IQM Garnet (Crystal 20). A robust empirical invariant is observed: π ν_ZZ τ* ≈ const ≈ 0.873 rad, confirmed across five independent IBM configurations (π ν_ZZ τ* = 0.873 ± 0.001 rad). A preliminary τ-scan on IQM Garnet shows a GPS sign reversal consistent with a zero-crossing near φ = 0.873 rad. The result establishes a robust empirical invariant within superconducting architectures, enabling feedforward calibration without full phase tomography. Related work:- EEDT Protocol: https://doi.org/10.5281/zenodo.19029431- Bell Decay Clock: https://doi.org/10.5281/zenodo.19367565- Software (Quantum-6502): https://doi.org/10.5281/zenodo.19160835
We derive analytically that the optimal feedforward phase for ZZ-coupling correction in superconducting qubits is given by the argument of the first circular moment of the accumulated phase distribution: φ* = arctan(S/C), where C = ⟨cos φ⟩ and S = ⟨sin φ⟩. A negative second derivative proves this is always a maximum. We measure GPS = P_ff − P_ref on IBM Heron r2 processors (ibm_kingston, ibm_marrakesh; five qubit pairs) and IQM Garnet (Crystal 20). A robust empirical invariant is observed: π ν_ZZ τ* ≈ const ≈ 0.873 rad, confirmed across five independent IBM configurations (π ν_ZZ τ* = 0.873 ± 0.001 rad). A preliminary τ-scan on IQM Garnet shows a GPS sign reversal consistent with a zero-crossing near φ = 0.873 rad. The result establishes a robust empirical invariant within superconducting architectures, enabling feedforward calibration without full phase tomography. Related work:- EEDT Protocol: https://doi.org/10.5281/zenodo.19029431- Bell Decay Clock: https://doi.org/10.5281/zenodo.19367565- Software (Quantum-6502): https://doi.org/10.5281/zenodo.19160835
Phase accumulation during idle intervals in superconducting qubits is well known, and virtual Z (Rz) gates are routinely used for correction. However, the distinction between unconditional (static) and conditional (timing-aware) phase correction has received limited experimental attention. Here we present experimental results on IBM Quantum superconducting processors showing that unconditional phase correction can systematically degrade fidelity in timing-misaligned regimes, specifically for Bell-type superposition under ZZ-dominated evolution. On ibm_kingston (Heron r2), unconditional correction yields GPS = -0.201 ± 0.007 (z = -27.9σ) while conditional MCM feedforward achieves GPS = +0.345 ± 0.007 (z = +47.96σ), a gap of ΔGPS = +0.546. GPS = F_ff - F_ref isolates the incremental effect of correction by removing baseline decoherence. This result does not invalidate static phase correction globally; the effect emerges only under timing-misaligned stochastic phase regimes. The physical mechanism is derived from H = ξ_ZZ|11⟩⟨11|: applying Rz(-φ*) unconditionally to one qubit of a Bell pair adds φ* to the relative phase rather than canceling it, doubling the phase error at τ*. Density matrix simulation with Kraus operators confirms GPS_static = -0.39 (model). These results demonstrate that phase correction effectiveness is strongly regime-dependent, and identify conditional phase correction as a synchronization primitive between stochastic phase evolution and control application. AI assistance (Claude, Anthropic) disclosed.Data: IBM Quantum Cloud, 2026-03-22, job d6vqgtov5rlc73f5a7kg.Code: https://github.com/okudat9/Quantum-6502
AbstractWe present Quantum-6502, a self-calibrating BIOS layer for Entanglement-Enhanced Dynamical Tracking (EEDT) based phase-storage on IBM Heron r2 superconducting processors. EEDT is a proof-of-concept phase-storage protocol that demonstrates how always-on ZZ coupling can be converted into a feedforward resource via mid-circuit measurement (MCM), rather than suppressed. Key Features of Quantum-6502 BIOS:Universal Pair Discovery (Layer 0): Automatically discovers suitable qubit pairs from backend calibration data using zero quantum jobs.Ultra-Light Calibration (ULC): Measures νZZ and T2 using only 2 jobs (9 circuits).Adaptive Execution: Executes QFEED+ adaptive feedforward to compensate for per-session drift without user intervention. Hardware Validation Results:Static vs. Dynamic Contrast: On ibm_kingston (Q149+Q150), the system yielded GPS_dynamic = +0.345 ± 0.007 versus GPS_static = −0.201 (Δ = +0.546, z = 47.96σ), assuming shot-level independence with binomial standard error. This contrast provides strong direct evidence consistent with the stochastic collapse-timing mechanism of EEDT.Cross-Backend Consistency: Validation on ibm_marrakesh confirms GPS = +0.037 (z = 8.5σ combined, 8 Holm-corrected conditions) and GPS = +0.726 (z = 66σ).Expanded Operating Regime: The kingston result extends the confirmed operating regime to ωZZT2 = 21.9. Based on the data, τ*/TMCM ≥ 5 is tentatively proposed as the operative constraint, ensuring that measurement overhead remains subdominant to coherent evolution.Robustness: Sensitivity analysis confirms that EEDT gain remains positive (GPS > 0) across a ±30% variance (covering observed drift ranges) from the MANUAL calibration estimate of νZZ. Consistency with AUTO-mode estimates (where available) indicates no systematic bias introduced by the MANUAL override. AvailabilityThe present validation is restricted to IBM Heron r2. All system code, raw experimental data, and IBM Quantum job IDs are publicly available in the linked repository.
Abstract Dynamic circuits with mid-circuit measurement (MCM) and classical feed-forward are essential building blocks for quantum error correction and adaptive quantum protocols. However, in superconducting quantum processors the readout process can induce coherent crosstalk on non-measured (spectator) qubits through the readout resonator, residual photons, and other hardware pathways, leading to deterministic phase kicks (effective Z rotations) that accumulate with the number of MCM events. In this work we present a minimal, reproducible two-qubit Ramsey diagnostic to quantify the spectator-induced phase shift as a function of the number of inserted mid-circuit measurements M, and we demonstrate that a simple virtual-Z (frame) compensation proportional to M can mitigate the resulting phase error on real hardware. In a high-SNR dataset archived on Zenodo (T = 40 μs, 4096 shots per circuit), the effective phase difference Δθeff(M) = θMEAS(M) - θDELAY(M) shifts monotonically in the negative direction as M increases, reaching Δθeff (M=8) = -1.944 ± 0.033 rad. While the contrast r decreases mildly with M, it remains relatively high (r ≈ 0.84 at M = 8), consistent with the effect appearing predominantly as a coherent phase rotation. To mitigate the error, we insert a virtual-Z rotation on the target qubit of the form Rz (comp × M) in the MEAS condition and scan the compensation coefficient comp (rad per measurement). In a MAP dataset (ibm_marrakesh, shots = 512, repeats = 1, M ∈ {0,1,2,4,8}, comp ∈ {-0.02,-0.01,0,+0.01,+0.02}), the RMS of the baseline-referenced phase error Δθeff(M) for M>0 is minimized at comp = -0.01, decreasing from 0.1393 rad (baseline) to 0.0601 rad. Finally, we implement a low-cost calibration probe (M = {0,8}) scanning five compensation settings comp ∈ {-0.02,-0.01,0,+0.01,+0.02} (1024 shots per circuit, repeats = 2) and estimate a session-dependent coefficient compopt ≈ -0.013 rad/measurement from a linear fit of Δθeff(M=8) versus comp. A same-session production sweep using comp = -0.01453 rad/measurement (close to compopt) shows a reduced residual phase error, supporting a practical probe → production workflow for dynamic-circuit execution. All analysis scripts and summary data are made publicly available to support independent reproduction of the reported phase metrics and figures.
Abstract Dynamic circuits with mid-circuit measurement (MCM) and classical feed-forward are essential building blocks for quantum error correction and adaptive quantum protocols. However, in superconducting quantum processors the readout process can induce coherent crosstalk on non-measured (spectator) qubits through the readout resonator, residual photons, and other hardware pathways, leading to deterministic phase kicks (effective Z rotations) that accumulate with the number of MCM events. In this work we present a minimal, reproducible two-qubit Ramsey diagnostic to quantify the spectator-induced phase shift as a function of the number of inserted mid-circuit measurements M, and we demonstrate that a simple virtual-Z (frame) compensation proportional to M can mitigate the resulting phase error on real hardware. In a high-SNR dataset archived on Zenodo (T = 40 μs, 4096 shots per circuit), the effective phase difference Δθeff(M) = θMEAS(M) - θDELAY(M) shifts monotonically in the negative direction as M increases, reaching Δθeff (M=8) = -1.944 ± 0.033 rad. While the contrast r decreases mildly with M, it remains relatively high (r ≈ 0.84 at M = 8), consistent with the effect appearing predominantly as a coherent phase rotation. To mitigate the error, we insert a virtual-Z rotation on the target qubit of the form Rz (comp × M) in the MEAS condition and scan the compensation coefficient comp (rad per measurement). In a MAP dataset (ibm_marrakesh, shots = 512, repeats = 1, M ∈ {0,1,2,4,8}, comp ∈ {-0.02,-0.01,0,+0.01,+0.02}), the RMS of the baseline-referenced phase error Δθeff(M) for M>0 is minimized at comp = -0.01, decreasing from 0.1393 rad (baseline) to 0.0601 rad. Finally, we implement a low-cost calibration probe (M = {0,8}) scanning five compensation settings comp ∈{-0.02,-0.01,0,+0.01,+0.02} (1024 shots per circuit, repeats = 2) and estimate a session-dependent coefficient compopt ≈ -0.013 rad/measurement from a linear fit of Δθeff(M=8) versus comp. A same-session production sweep using comp = -0.01453 rad/measurement (close to compopt) shows a reduced residual phase error, supporting a practical probe → production workflow for dynamic-circuit execution.
Dynamic circuits with mid-circuit measurement (MCM) and classical feed-forward are essential building blocks for quantum error correction and adaptive quantum protocols. However, in superconducting quantum processors the readout process can induce coherent crosstalk on non-measured (spectator) qubits through the readout resonator, residual photons, and other hardware pathways, leading to deterministic phase kicks (effective Z rotations) that accumulate with the number of MCM events. In this work we present a minimal, reproducible two-qubit Ramsey diagnostic to quantify the spectator-induced phase shift as a function of the number of inserted mid-circuit measurements M, and we demonstrate that a simple virtual-Z (frame) compensation proportional to M can mitigate the resulting phase error on real hardware. In a high-SNR dataset archived on Zenodo (T = 40 μs, 4096 shots per circuit), the effective phase difference Δθeff(M) = θMEAS(M) - θDELAY(M) shifts monotonically in the negative direction as M increases, reaching Δθeff (M=8) = -1.944 ± 0.033 rad. While the contrast r decreases mildly with M, it remains relatively high (r ≈ 0.84 at M = 8), consistent with the effect appearing predominantly as a coherent phase rotation. To mitigate the error, we insert a virtual-Z rotation on the target qubit of the form Rz (comp × M) in the MEAS condition and scan the compensation coefficient comp (rad per measurement). In a MAP dataset (ibm_marrakesh, shots = 512, repeats = 1, M ∈ {0,1,2,4,8}, comp ∈ {-0.02,-0.01,0,+0.01,+0.02}), the RMS of the baseline-referenced phase error DθCT(M) for M>0 is reduced from 0.1393 rad (baseline) to 0.0601 rad at comp = -0.01, demonstrating effective error reduction with a single-parameter circuit-level correction. Finally, using a pay-as-you-go execution environment we run a low-cost “calibration probe” (M = {0,8}, three comp points, 32 circuits total) and obtain compopt ≈ +0.0088 rad/measurement from a three-point linear fit. This supports a practical two-stage workflow in which a short probe estimates the session-dependent compensation coefficient, followed by same-session production runs with the calibrated virtual-Z correction.
Universal Fault Tolerance remains the grand challenge of quantum computing. In this work, we present empirical evidence for a passive error suppression protocol based on a topological geometric phase constraint, Y = π²/φ ≈ 6.0998 rad. Through controlled A/B testing on IBM Quantum processors (127-qubit 'Eagle' and 133-qubit 'Heron'), we report a consistent fidelity improvement of ~ 7% in the generation of Bell States compared to standard circuit implementations. These findings suggest that topological phase engineering acts as a "Zero-Overhead" stabilization layer suitable for NISQ-era hardware.