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.
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.
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.
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.
Complex-Time (CT) theory posits that physical time is not exhausted by a single real parameter t, but includes an internal rotation phase φ such that the time coordinate is τ = t + iφ. In this ontology, φ is not a mere mathematical artifact of complex amplitudes: it is a physical degree of freedom that mediates a rotation between two logically distinct time aspects—future-pending (fp) time that retains unrealized alternatives, and past-fixed (pf) time that corresponds to realized events and stable records. Dynamic quantum circuits with mid-circuit measurement (MCM) are an unusually direct laboratory for CT because they contain repeated, addressable fp→pf conversion events *inside a single circuit*.Here we analyze IBM superconducting quantum-processor data and show that the target-qubit equatorial coherence z ≡ ⟨σx⟩ + i⟨σy⟩ undergoes a deterministic complex rotation whose angle is set primarily by the number M of spectator mid-circuit measurements. In Dataset A (4096 shots; 5 repeats), the complex ratio w(M) ≡ z_MEAS(M)/z_DELAY(M) displays an event-additive phase shift Δθ_eff(M) = arg w(M) that grows from ≈ 0 at M = 0–1 to Δθ_eff(M=8) ≈ 2.279 rad, with an average increment of δ ≈ 0.323 rad per added measurement. In Dataset B (1024 shots), inserting a virtual-Z compensation Rz(comp×M) on the target yields linear response d(Δθ_eff)/d(comp) ≈ M and demonstrates coherent controllability of the rotation.To probe whether the effect is reducible to schedule-level artifacts alone, Dataset C (CT3 Layer3) uses identical classical control-flow (if-structure) while changing only the spectator’s internal phase coherence: mode A forces |+⟩ before each mid-measurement, whereas mode B allows a |+⟩/|−⟩ mixture to accumulate. A small but systematic phase difference Δθ_AB(M) = θ_A(M) − θ_B(M) emerges (≈ 0.06 rad at M = 4–8) alongside a large change in spectator ⟨σx⟩, consistent with an operational coupling between the driven phase rotation and an internal i-phase degree of freedom.We interpret these results as operational evidence that an i-axis phase coordinate is physically actionable: it can be incremented by pf-creating events and steered by coherent frame updates. This provides a concrete, hardware-grounded route for testing CT claims about the physical reality of the imaginary-time phase. Keywords: complex time; τ = t + iφ; i-phase; past-fixed / future-pending; dynamic circuits; mid-circuit measurement; complex-plane diagnostics; virtual-Z compensation.
Mid-circuit measurement (MCM) enables qubit reuse and dynamic control in quantum processors, underpinning resource-efficient algorithms and quantum error correction. However, MCM introduces non-negligible errors, including measurement-induced crosstalk, idling-qubit decoherence, and reset infidelity. Additionally, MCM error rates vary across qubits within the same processor, with some exhibiting substantially higher errors. Existing compilers, including Qiskit-compiler and QR-Map, ignore MCM error variations, leading to fidelity loss in circuits with frequent MCM operations.
In this paper, we present MERA, an MCM-error-aware quantum compilation framework that integrates MCM error modeling into the layout, routing, and scheduling stages. We first use the lightweight profiling method to capture a per-qubit MCM error distribution that remains stable for over 24 hours. The distribution guides MERA in applying MCM-error-aware layout and routing, while MERA further incorporates context-aware dynamic decoupling to improve fidelity. In evaluation, MERA delivers 24.94% -- 80.48% average fidelity gains over the Qiskit-compiler (optimization level 3) across different groups in selected benchmarks without additional SWAP overhead. It also boosts the fidelity of QR-Map-generated circuits by 29.26% on average (up to 122.58%), demonstrating its effectiveness for circuits with frequent MCM operations.
Abstract Complex-Time (CT) theory posits that physical time is not exhausted by a single real parameter t, but includes an internal rotation phase φ such that the time coordinate is τ = t + iφ. In this ontology, φ is not a mere mathematical artifact of complex amplitudes: it is a physical degree of freedom that mediates a rotation between two logically distinct time aspects—future-pending (fp) time that retains unrealized alternatives, and past-fixed (pf) time that corresponds to realized events and stable records. Dynamic quantum circuits with mid-circuit measurement (MCM) are an unusually direct laboratory for CT because they contain repeated, addressable fp→pf conversion events *inside a single circuit*.Here we analyze IBM superconducting quantum-processor data and show that the target-qubit equatorial coherence z ≡ ⟨σx⟩ + i⟨σy⟩ undergoes a deterministic complex rotation whose angle is set primarily by the number M of spectator mid-circuit measurements. In Dataset A (4096 shots; 5 repeats), the complex ratio w(M) ≡ z_MEAS(M)/z_DELAY(M) displays an event-additive phase shift Δθ_eff(M) = arg w(M) that grows from ≈ 0 at M = 0–1 to Δθ_eff(M=8) ≈ 2.279 rad, with an average increment of δ ≈ 0.323 rad per added measurement. In Dataset B (1024 shots), inserting a virtual-Z compensation Rz(comp×M) on the target yields linear response d(Δθ_eff)/d(comp) ≈ M and demonstrates coherent controllability of the rotation.To probe whether the effect is reducible to schedule-level artifacts alone, Dataset C (CT3 Layer3) uses identical classical control-flow (if-structure) while changing only the spectator’s internal phase coherence: mode A forces |+⟩ before each mid-measurement, whereas mode B allows a |+⟩/|−⟩ mixture to accumulate. A small but systematic phase difference Δθ_AB(M) = θ_A(M) − θ_B(M) emerges (≈ 0.06 rad at M = 4–8) alongside a large change in spectator ⟨σx⟩, consistent with an operational coupling between the driven phase rotation and an internal i-phase degree of freedom.We interpret these results as operational evidence that an i-axis phase coordinate is physically actionable: it can be incremented by pf-creating events and steered by coherent frame updates. This provides a concrete, hardware-grounded route for testing CT claims about the physical reality of the imaginary-time phase. Keywords: complex time; τ = t + iφ; i-phase; past-fixed / future-pending; dynamic circuits; mid-circuit measurement; complex-plane diagnostics; virtual-Z compensation.