Spectator Phase Shift Scaling with Mid-Circuit Measurement Count and Its Mitigation via Virtual-Z Compensation on a Superconducting Quantum Processor - A Minimal Ramsey Diagnostic on IBM Quantum ibm_marrakesh (T = 40 μs) -

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.