Complex-plane diagnostic for MCM-induced coherent phase errors: w(M) and log w(M)

This record timestamps a compact complex-plane diagnostic for coherent phase errors induced by mid-circuit measurement (MCM) in superconducting dynamic circuits. Motivation: MCM is an essential primitive for dynamic circuits (e.g., QEC and adaptive protocols), yet on superconducting processors the readout operation can be invasive and can induce correlated, deterministic phase “kicks” (effective Z rotations) on non-measured qubits. Because coherent phase errors are, in principle, correctable by low-overhead frame updates (virtual-Z), it is useful to have a diagnostic that separates coherent rotation from incoherent shrinkage at a glance. Key definitions (Ramsey-type two-qubit diagnostic):- Complex equatorial signal: z ≡ ⟨σx⟩ + i⟨σy⟩ ≡ r e^{iθ}- Matched control pair: MEAS (insert M MCM events on a spectator) vs DELAY (insert matched delays)- Complex ratio: w(M) ≡ z_MEAS(M) / z_DELAY(M)- Decomposition: log w(M) = ln|w(M)| + i arg(w(M)) Interpretation:- arg(w(M)) gives the relative phase rotation: arg(w(M)) = θ_MEAS(M) − θ_DELAY(M) ≡ Δθ_eff(M)- ln|w(M)| captures relative shrinkage (contrast loss): ln|w(M)| = ln[r_MEAS(M)/r_DELAY(M)]Thus, on the Argand plane the angle of w(M) visualizes coherent rotation, while the radius |w(M)| visualizes relative shrinkage. Contents of this record:- A short note (v0) and figures demonstrating w(M) and log w(M) using high-SNR summary values (Table 2) from an existing archived dataset (T = 40 μs; 4096 shots/circuit). This v0 demonstration is intentionally “Table-derived” and is provided to timestamp the diagnostic representation and its interpretation. Planned update:- An independent confirmation dataset (“A-lite core”) will be acquired in a short QPU window (M ∈ {0,1,2,4,8} × MEAS/DELAY × X/Y; 1024 shots; 20 circuits total) and will replace the present Table-derived demonstration in a subsequent Zenodo version. Related record:- Underlying dataset and analysis materials are archived on Zenodo (DOI: 10.5281/zenodo.18288058). Acknowledgment:This work used the IBM Quantum service. The views expressed are those of the author and do not reflect the official policy or position of IBM or the IBM Quantum team.