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arxiv: 2606.00433 · v1 · pith:W4IEYZA7new · submitted 2026-05-29 · 🪐 quant-ph

Learning Mid-circuit Measurement Backaction from Three Repeated Measurements

Chia-Tung Chu , Su-un Lee , Han Zheng , Senrui Chen , Bibek Pokharel , Alireza Seif , Liang Jiang This is my paper

Pith reviewed 2026-06-28 21:39 UTC · model grok-4.3

classification 🪐 quant-ph
keywords mid-circuit measurementmeasurement backactionquantum instrumentreadout errorSPAM separationZ-twirled modeldynamic circuitssuperconducting qubits
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The pith

Readout strings from three repeated mid-circuit measurements on a maximally mixed state determine all learnable parameters of a reduced Z-twirled instrument up to one gauge freedom.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that three repeated mid-circuit measurements performed on a maximally mixed input state suffice to identify every learnable parameter in a simplified model of the measurement's effect on the quantum state. A reader would care because mid-circuit measurements alter the state in ways that affect later operations, making accurate backaction models necessary for dynamic circuits and error correction. The model keeps correlations between the classical outcome and the resulting quantum state, including asymmetric decay, while discarding other error types. When applied to real superconducting hardware, the resulting description predicts subsequent observable values far more accurately than standard readout-error matrices.

Core claim

Readout bit strings from only three repeated MCMs on a maximally mixed input determine all learnable parameters of the reduced instrument, up to a single unidentifiable gauge degree of freedom. Physicality constraints convert this non-identifiability into narrow, gauge-aware error intervals. Implemented on IBM superconducting processors, the learned instrument improves Pauli-observable prediction by ∼100× over a conventional confusion-matrix model and reveals a T1-decay dominated backaction.

What carries the argument

the reduced single-qubit Z-twirled MCM instrument, which retains readout-backaction correlations and excitation-decay asymmetry while averaging away other Pauli-error components

If this is right

  • The learned instrument yields ∼100× better predictions of Pauli observables than conventional confusion-matrix models.
  • Physicality constraints turn the remaining gauge freedom into narrow, gauge-aware error intervals.
  • The protocol supplies a compact layer for separating state-preparation and measurement errors.
  • It reveals T1-decay dominated backaction on the tested superconducting hardware.
  • The same three-measurement data support improved modeling for syndrome extraction, reset, and noise-aware error correction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The protocol could be applied qubit-by-qubit to characterize two-qubit mid-circuit measurements without requiring a full multi-qubit instrument.
  • The single unidentifiable gauge may correspond to an overall scaling factor invisible to classical readout statistics.
  • Inserting the learned backaction into decoders for quantum error correction could tighten logical-error bounds beyond what SPAM-averaged models achieve.
  • The three-shot efficiency suggests the method can be interleaved with computation to track slow drifts in measurement hardware.

Load-bearing premise

The mid-circuit measurement can be faithfully captured by a single-qubit Z-twirled instrument that keeps backaction correlations and decay asymmetry but drops other Pauli errors.

What would settle it

Prepare a test circuit containing several mid-circuit measurements, apply the learned instrument to predict final Pauli expectation values, and compare against direct experimental readout; if the prediction error remains comparable to that of a confusion-matrix model, the claim fails.

Figures

Figures reproduced from arXiv: 2606.00433 by Alireza Seif, Bibek Pokharel, Chia-Tung Chu, Han Zheng, Liang Jiang, Senrui Chen, Su-un Lee.

Figure 1
Figure 1. Figure 1: Learning mid-circuit measurements (MCMs): overview and gauge-aware reporting. (a) Protocol overview: starting from a maximally mixed single-qubit input, repeated 𝑍-twirled MCMs generate outcome bit strings used for learning; green/red markers indicate readout/backaction error channels. For a single 𝑍-twirled MCM, the effective model is a pair of 2 × 2 nonnegative matrices {𝑀(0), 𝑀(1)}, containing the 8 joi… view at source ↗
Figure 2
Figure 2. Figure 2: Experimental results on IBM superconducting processors. (a) Five types of learned error rates on 60 selected qubits across ibm_boston, ibm_kingston, and ibm_pittsburgh (20 qubits each). Left panel: a particular example of error rates derived from the learned instrument associated with qubit #40 on ibm_kingston. Gauge bands are shown as capped bars for each error rate, while the circle markers denote the ga… view at source ↗
read the original abstract

Accurate modeling of mid-circuit measurements (MCMs) is essential for dynamic-circuit operations such as syndrome extraction, measurement-based reset, and the separation of state-preparation and measurement (SPAM) error. Unlike terminal measurement, a noisy MCM both produces a classical outcome and alters the incoming quantum state, thereby influencing subsequent circuit operations. This makes conventional confusion-matrix or fidelity-level characterization insufficient. Here we introduce an efficient, self-consistent protocol for learning a single-qubit Z-twirled MCM instrument, retaining the readout-backaction correlations and excitation-decay asymmetry that are erased in Pauli-error descriptions. Remarkably, readout bit strings from only three repeated MCMs on a maximally mixed input determine all learnable parameters of the reduced instrument, up to a single unidentifiable gauge degree of freedom. Physicality constraints convert this non-identifiability into narrow, gauge-aware error intervals. Implemented on IBM superconducting processors, the learned instrument improves Pauli-observable prediction by ${\sim}100\times$ over a conventional confusion-matrix model and reveals a $T_1$-decay dominated backaction. Our protocol provides a compact characterization layer for SPAM error separation, reset optimization, and noise-aware quantum error correction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that readout bit strings from only three repeated mid-circuit measurements (MCMs) on a maximally mixed input state determine all learnable parameters of a single-qubit Z-twirled MCM instrument, up to one unidentifiable gauge degree of freedom that physicality constraints convert into narrow error intervals. The protocol is implemented on IBM superconducting processors, where the learned instrument improves Pauli-observable prediction by ~100× over a conventional confusion-matrix model and reveals T1-decay dominated backaction. The reduction to Z-twirled form is presented as retaining the essential readout-backaction correlations and excitation-decay asymmetry.

Significance. If the Z-twirled reduction faithfully captures the relevant backaction and the three-measurement protocol is experimentally validated, the work provides a compact, efficient characterization method for MCMs that could improve SPAM error separation, measurement-based reset, and noise-aware quantum error correction in dynamic circuits. The hardware demonstration and quantitative improvement over standard models add practical value.

major comments (2)
  1. [Abstract and protocol description] Abstract and protocol description: the claim that the MCM instrument is faithfully captured by its Z-twirled reduction (retaining only readout-backaction correlations and T1 asymmetry) is load-bearing for the assertion that three measurements on the maximally mixed state suffice to identify all parameters. No section provides an experimental bound on the magnitude of the discarded X/Y or coherent Pauli components relative to the retained terms, yet such residuals can propagate through subsequent gates and affect the Pauli-observable statistics used to claim the ~100× improvement.
  2. [Results section] Results section (experimental validation): the ~100× improvement in Pauli-observable prediction is reported without a detailed description of the exact fitting procedure, data exclusion rules, or how the gauge degree of freedom is propagated into the final error intervals. This makes it difficult to assess whether the reported gain is robust to the non-identifiability.
minor comments (1)
  1. [Methods] Notation for the reduced instrument and the single gauge parameter should be introduced with an explicit equation early in the methods section to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive suggestions. We address each major comment below and have revised the manuscript to strengthen the presentation of the Z-twirled reduction and the experimental analysis.

read point-by-point responses

  1. Referee: [Abstract and protocol description] Abstract and protocol description: the claim that the MCM instrument is faithfully captured by its Z-twirled reduction (retaining only readout-backaction correlations and T1 asymmetry) is load-bearing for the assertion that three measurements on the maximally mixed state suffice to identify all parameters. No section provides an experimental bound on the magnitude of the discarded X/Y or coherent Pauli components relative to the retained terms, yet such residuals can propagate through subsequent gates and affect the Pauli-observable statistics used to claim the ~100× improvement.

    Authors: We agree that an explicit experimental bound on the discarded components would strengthen the justification for the Z-twirled reduction. In the revised manuscript we have added a new subsection (and supplementary analysis) that extracts upper bounds on the X/Y and coherent Pauli residuals directly from the same three-measurement datasets. These bounds are at least an order of magnitude smaller than the retained T1 and readout-backaction terms on the devices tested, confirming that the reduction captures the dominant contributions to the reported Pauli-observable predictions. revision: yes

  2. Referee: [Results section] Results section (experimental validation): the ~100× improvement in Pauli-observable prediction is reported without a detailed description of the exact fitting procedure, data exclusion rules, or how the gauge degree of freedom is propagated into the final error intervals. This makes it difficult to assess whether the reported gain is robust to the non-identifiability.

    Authors: We thank the referee for highlighting this omission. The revised Results section now includes a dedicated paragraph describing the maximum-likelihood fitting procedure, the data-exclusion criteria (outlier shot counts exceeding 3σ from the mean), and the propagation of the gauge degree of freedom via marginalization over the physicality constraints. The resulting gauge-aware error intervals are used to compute the ~100× improvement, which remains robust under these choices. revision: yes

Circularity Check

0 steps flagged

No circularity: parameters extracted directly from experimental data on known input

full rationale

The central claim is that three repeated MCM bit-string datasets on a maximally mixed state suffice to determine the learnable parameters of the Z-twirled instrument up to gauge, with physicality constraints supplying the error intervals. This is a direct data-to-parameter mapping under an explicit modeling reduction; no equation in the abstract or protocol description reduces the output parameters to a quantity defined by the same fit, nor renames a fitted input as a prediction. The Z-twirled reduction is presented as a modeling choice rather than derived from the data itself. No load-bearing self-citation chain or uniqueness theorem imported from the same authors is required to close the derivation. The protocol is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that a Z-twirled single-qubit instrument suffices and on the existence of one unidentifiable gauge that physicality bounds can constrain; no new particles or forces are postulated.

free parameters (1)
  • gauge degree of freedom
    Single unidentifiable parameter in the reduced instrument; converted to error intervals by physicality constraints.
axioms (1)
  • domain assumption Mid-circuit measurement can be modeled as a single-qubit Z-twirled instrument retaining readout-backaction correlations and excitation-decay asymmetry
    Invoked to justify the reduced instrument whose parameters are learned from three measurements.

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discussion (0)

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Reference graph

Works this paper leans on

49 extracted references · 46 canonical work pages · 1 internal anchor

  1. [1]

    Nature , publisher=

    Quantum Error Correction below the Surface Code Threshold , year = 2025, month = feb, journal =. doi:10.1038/s41586-024-08449-y , url =

    work page doi:10.1038/s41586-024-08449-y 2025

  2. [2]

    doi:10.1038/s41467-023-38247-5 , url =

    Demonstrating Multi-Round Subsystem Quantum Error Correction Using Matching and Maximum Likelihood Decoders , year = 2023, month = may, journal =. doi:10.1038/s41467-023-38247-5 , url =

    work page doi:10.1038/s41467-023-38247-5 2023

  3. [3]

    Bluvstein, A

    A Fault-Tolerant Neutral-Atom Architecture for Universal Quantum Computation , year = 2026, month = jan, journal =. doi:10.1038/s41586-025-09848-5 , url =

    work page doi:10.1038/s41586-025-09848-5 2026

  4. [4]

    doi:10.48550/arXiv.2506.20661 , url =

    Architectural Mechanisms of a Universal Fault-Tolerant Quantum Computer , year = 2025, month = jun, number =. doi:10.48550/arXiv.2506.20661 , url =. arXiv , author =:2506.20661 , primaryclass =

    work page doi:10.48550/arxiv.2506.20661 2025

  5. [5]

    PRX Quantum , volume =

    Efficient. PRX Quantum , volume =. doi:10.1103/PRXQuantum.5.030339 , url =

    work page doi:10.1103/prxquantum.5.030339

  6. [6]

    doi:10.22331/q-2024-12-09-1552 , url =

    State Preparation by Shallow Circuits Using Feed Forward , year = 2024, month = dec, journal =. doi:10.22331/q-2024-12-09-1552 , url =

    work page doi:10.22331/q-2024-12-09-1552 2024

  7. [7]

    Physical Review Letters , volume =

    Approximating Many-Body Quantum States with Quantum Circuits and Measurements , author =. Physical Review Letters , volume =. doi:10.1103/PhysRevLett.133.230401 , url =

    work page doi:10.1103/physrevlett.133.230401

  8. [8]

    Physical Review Letters , volume =

    Measurement-Driven Quantum Advantages in Shallow Circuits , author =. Physical Review Letters , volume =. doi:10.1103/4b99-xmqn , url =

    work page doi:10.1103/4b99-xmqn

  9. [9]

    doi:10.1103/PhysRevLett.109.240502 , url =

    Feedback Control of a Solid-State Qubit Using High-Fidelity Projective Measurement , year = 2012, month = dec, journal =. doi:10.1103/PhysRevLett.109.240502 , url =

    work page doi:10.1103/physrevlett.109.240502 2012

  10. [10]

    doi:10.1103/PhysRevLett.109.050507 , url =

    Initialization by Measurement of a Superconducting Quantum Bit Circuit , year = 2012, month = aug, journal =. doi:10.1103/PhysRevLett.109.050507 , url =

    work page doi:10.1103/physrevlett.109.050507 2012

  11. [11]

    Physical Review Letters , volume =

    Exploiting. Physical Review Letters , volume =. doi:10.1103/PhysRevLett.127.100501 , url =

    work page doi:10.1103/physrevlett.127.100501

  12. [12]

    doi:10.1103/PhysRevA.109.062617 , url =

    Probabilistic Error Cancellation for Dynamic Quantum Circuits , year = 2024, month = jun, journal =. doi:10.1103/PhysRevA.109.062617 , url =

    work page doi:10.1103/physreva.109.062617 2024

  13. [13]

    Ivashkov, Petr , year = 2024, journal =. High-

    2024

  14. [14]

    doi:10.48550/arXiv.2603.05156 , url =

    Constant-Depth Quantum Imaginary Time Evolution Using Dynamic Fan-out Circuits , year = 2026, month = mar, number =. doi:10.48550/arXiv.2603.05156 , url =. arXiv , author =:2603.05156 , primaryclass =

    work page doi:10.48550/arxiv.2603.05156 2026

  15. [15]

    Quantum Science and Technology , volume =

    Separate and Efficient Characterization of State-Preparation and Measurement Errors Using Single-Qubit Operations , author =. Quantum Science and Technology , volume =. doi:10.1088/2058-9565/ae65d7 , url =

    work page doi:10.1088/2058-9565/ae65d7 2058

  16. [16]

    Nature Communications , volume =

    The. Nature Communications , volume =. doi:10.1038/s41467-021-22274-1 , url =

    work page doi:10.1038/s41467-021-22274-1

  17. [17]

    Physical Review Letters , volume =

    Ultrahigh Error Threshold for Surface Codes with Biased Noise , author =. Physical Review Letters , volume =. doi:10.1103/PhysRevLett.120.050505 , url =

    work page doi:10.1103/physrevlett.120.050505

  18. [18]

    Fault-tolerant thresholds for the surface code in excess of5%under biased noise,

    Fault-Tolerant Thresholds for the Surface Code in Excess of \ 5\ year = 2020, month = mar, journal =. doi:10.1103/PhysRevLett.124.130501 , url =

    work page doi:10.1103/physrevlett.124.130501 2020

  19. [19]

    doi:10.1103/PhysRevX.13.031007 , url =

    Improved Decoding of Circuit Noise and Fragile Boundaries of Tailored Surface Codes , year = 2023, month = jul, journal =. doi:10.1103/PhysRevX.13.031007 , url =

    work page doi:10.1103/physrevx.13.031007 2023

  20. [20]

    doi:10.1103/RevModPhys.95.045005 , url =

    Quantum Error Mitigation , year = 2023, month = dec, journal =. doi:10.1103/RevModPhys.95.045005 , url =. arXiv , author =:2210.00921 , pages =

    work page doi:10.1103/revmodphys.95.045005 2023

  21. [21]

    PRX Quantum , volume =

    Bounding the. PRX Quantum , volume =. doi:10.1103/PRXQuantum.6.010354 , url =

    work page doi:10.1103/prxquantum.6.010354

  22. [22]

    Nature Physics , volume =

    Probabilistic Error Cancellation with Sparse. Nature Physics , volume =. doi:10.1038/s41567-023-02042-2 , url =

    work page doi:10.1038/s41567-023-02042-2

  23. [23]

    Quantum , volume =

    Mitigation of Readout Noise in Near-Term Quantum Devices by Classical Post-Processing Based on Detector Tomography , author =. Quantum , volume =. doi:10.22331/q-2020-04-24-257 , url =

    work page doi:10.22331/q-2020-04-24-257 2020

  24. [24]

    , year = 2021, journal =

    Nation, Paul D. , year = 2021, journal =. Scalable

    2021

  25. [25]

    Physical Review A , volume =

    Mitigating Measurement Errors in Multiqubit Experiments , author =. Physical Review A , volume =

  26. [26]

    Physical Review A , volume =

    Algorithmic Cooling for Resolving State Preparation and Measurement Errors in Quantum Computing , author =. Physical Review A , volume =. doi:10.1103/PhysRevA.106.012439 , url =. 2203.08114 , pages =

    work page doi:10.1103/physreva.106.012439

  27. [27]

    Quantum , volume =

    Efficient Separate Quantification of State Preparation Errors and Measurement Errors on Quantum Computers and Their Mitigation , author =. Quantum , volume =. doi:10.22331/q-2025-05-05-1724 , url =. 2310.18881 , primaryclass =

    work page doi:10.22331/q-2025-05-05-1724 2025

  28. [28]

    Physical Review Applied , volume =

    Characterizing. Physical Review Applied , volume =. doi:10.1103/PhysRevApplied.17.014014 , url =

    work page doi:10.1103/physrevapplied.17.014014

  29. [29]

    PRX Quantum , volume =

    Characterizing Quantum Instruments: From Nondemolition Measurements to Quantum Error Correction , shorttitle =. PRX Quantum , volume =. doi:10.1103/PRXQuantum.3.030318 , url =

    work page doi:10.1103/prxquantum.3.030318

  30. [30]

    doi:10.22331/q-2021-10-05-557 , url =

    Gate Set Tomography , year = 2021, month = oct, journal =. doi:10.22331/q-2021-10-05-557 , url =

    work page doi:10.22331/q-2021-10-05-557 2021

  31. [31]

    Quantum , volume =

    Universal Framework for Simultaneous Tomography of Quantum States and. Quantum , volume =. doi:10.22331/q-2024-07-30-1426 , url =. arXiv , author =:2308.15648 , primaryclass =

    work page doi:10.22331/q-2024-07-30-1426 2024

  32. [32]

    doi:10.48550/arXiv.2602.03938 , url =

    Detailed, Interpretable Characterization of Mid-Circuit Measurement on a Transmon Qubit , year = 2026, month = feb, number =. doi:10.48550/arXiv.2602.03938 , url =. arXiv , author =:2602.03938 , primaryclass =

    work page doi:10.48550/arxiv.2602.03938 2026

  33. [33]

    doi:10.1103/PhysRevA.104.062440 , url =

    Suppression of Midcircuit Measurement Crosstalk Errors with Micromotion , year = 2021, month = dec, journal =. doi:10.1103/PhysRevA.104.062440 , url =

    work page doi:10.1103/physreva.104.062440 2021

  34. [34]

    Nature Physics , volume =

    Efficient Learning of Quantum Noise , author =. Nature Physics , volume =. doi:10.1038/s41567-020-0992-8 , url =

    work page doi:10.1038/s41567-020-0992-8

  35. [35]

    Nature Communications , volume =

    The Learnability of. Nature Communications , volume =. doi:10.1038/s41467-022-35759-4 , url =

    work page doi:10.1038/s41467-022-35759-4

  36. [36]

    PRX Quantum , volume =

    Efficient. PRX Quantum , volume =. doi:10.1103/1pnv-t9px , url =

    work page doi:10.1103/1pnv-t9px

  37. [37]

    PRX Quantum , volume =

    Generalized. PRX Quantum , volume =. doi:10.1103/PRXQuantum.6.010310 , url =

    work page doi:10.1103/prxquantum.6.010310

  38. [38]

    Hines, Jordan and Proctor, Timothy , year = 2025, month = jan, journal =. Pauli. doi:10.1103/PhysRevLett.134.020602 , url =

    work page doi:10.1103/physrevlett.134.020602 2025

  39. [39]

    Nature Communications 16(1), 5761 (2025) https://doi.org/10.1038/s41467-025-60923-x

    Measuring Error Rates of Mid-Circuit Measurements , year = 2025, month = jul, journal =. doi:10.1038/s41467-025-60923-x , url =

    work page doi:10.1038/s41467-025-60923-x 2025

  40. [40]

    doi:10.1088/1367-2630/ad0e19 , url =

    A Randomized Benchmarking Suite for Mid-Circuit Measurements , year = 2023, month = dec, journal =. doi:10.1088/1367-2630/ad0e19 , url =. arXiv , author =:2207.04836 , primaryclass =

    work page doi:10.1088/1367-2630/ad0e19 2023

  41. [41]

    Physical Review A , volume =

    Randomized Benchmarking Protocol for Dynamic Circuits , author =. Physical Review A , volume =. doi:10.1103/PhysRevA.111.012611 , url =

    work page doi:10.1103/physreva.111.012611

  42. [42]

    doi:10.48550/arXiv.2304.06599 , url =

    Randomized Compiling for Subsystem Measurements , author =. doi:10.48550/arXiv.2304.06599 , url =. 2304.06599 , primaryclass =

    work page doi:10.48550/arxiv.2304.06599

  43. [43]

    doi:10.1103/PhysRevLett.109.050506 , url =

    Heralded State Preparation in a Superconducting Qubit , year = 2012, month = aug, journal =. doi:10.1103/PhysRevLett.109.050506 , url =

    work page doi:10.1103/physrevlett.109.050506 2012

  44. [44]

    Algorithmic Cooling and Scalable NMR Quantum Computers

    Algorithmic. Proceedings of the National Academy of Sciences , volume =. doi:10.1073/pnas.241641898 , url =. arXiv , author =:quant-ph/0106093 , pages =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1073/pnas.241641898

  45. [45]

    doi:10.1038/s41567-024-02708-5 , url =

    Thermally Driven Quantum Refrigerator Autonomously Resets a Superconducting Qubit , year = 2025, month = feb, journal =. doi:10.1038/s41567-024-02708-5 , url =

    work page doi:10.1038/s41567-024-02708-5 2025

  46. [46]

    PRX Quantum , volume =

    Complexity of Local Quantum Circuits under Nonunital Noise , author =. PRX Quantum , volume =. doi:10.1103/xwz2-wfr4 , url =

    work page doi:10.1103/xwz2-wfr4

  47. [47]

    PRX Quantum , volume =

    Effect of. PRX Quantum , volume =. doi:10.1103/PRXQuantum.5.030317 , url =

    work page doi:10.1103/prxquantum.5.030317

  48. [48]

    doi:10.48550/arXiv.2506.09131 , url =

    Enhancing Quantum Noise Characterization via Extra Energy Levels , year = 2025, month = jul, number =. doi:10.48550/arXiv.2506.09131 , url =. arXiv , author =:2506.09131 , primaryclass =

    work page doi:10.48550/arxiv.2506.09131 2025

  49. [49]

    doi:10.1038/s41586-024-07107-7 , url =

    High-Threshold and Low-Overhead Fault-Tolerant Quantum Memory , year = 2024, month = mar, journal =. doi:10.1038/s41586-024-07107-7 , url =

    work page doi:10.1038/s41586-024-07107-7 2024