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arxiv: 2509.19448 · v1 · submitted 2025-09-23 · 🪐 quant-ph

Separate and efficient characterization of state-preparation and measurement errors using single-qubit operations

Muhammad Qasim Khan , Leigh M. Norris , Lorenza Viola This is my paper

Pith reviewed 2026-05-18 14:17 UTC · model grok-4.3

classification 🪐 quant-ph
keywords SPAM errorsstate preparationmeasurement errorsquantum error characterizationsingle-qubit operationsIBM Quantumerror mitigation
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The pith

A protocol using only single-qubit gates and repeated non-destructive measurements can separately characterize state-preparation and measurement errors with circuit depth independent of system size.

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

The paper presents a method to isolate and quantify errors that occur when preparing quantum states versus those that occur during measurement. In current hardware these SPAM errors often exceed gate errors, so separating them supports targeted hardware fixes and better mitigation. The protocol interleaves high-fidelity single-qubit gates with repeated measurements performed without resetting the qubit. Because the circuit depth does not grow with the number of qubits, the same sequence works in parallel across an entire device. Experiments on IBM Quantum hardware extracted SP infidelities up to 6.57 percent and readout errors up to 19.1 percent, while simulations confirmed that measurement-error mitigation alone produces biased observable values when state-preparation errors are ignored.

Core claim

We show how to construct a protocol that can efficiently and separately characterize the SP and M error parameters by using only high-fidelity single-qubit gates and repeated single-qubit measurements without reset. The measurements are assumed to be non-destructive and M errors are taken to be (spatially and temporally) uncorrelated and classical. Notably, the circuit depth of the protocol is independent of system size, and the target parameters may be characterized to a precision that is only limited by the number of experimental repetitions.

What carries the argument

A fixed-depth sequence of single-qubit gates interleaved with repeated non-destructive measurements that isolates state-preparation errors from measurement errors through their distinct signatures in the observed outcome statistics.

If this is right

  • SP and M errors can be extracted separately for many qubits at once without multi-qubit gates or resets.
  • Precision of the estimates is limited only by the number of experimental repetitions, not by circuit depth.
  • Accounting for state-preparation errors removes bias that remains when only measurement-error mitigation is applied.
  • The same fixed sequence works on devices of any size because depth does not scale with qubit number.

Where Pith is reading between the lines

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

  • The method could be embedded in routine calibration loops on large processors where full SPAM tomography becomes impractical.
  • Relaxing the classical uncorrelated measurement-error assumption would require additional sequences to detect temporal or spatial correlations.
  • Combining the extracted rates with gate-error data could produce more accurate overall error budgets for algorithm design.

Load-bearing premise

Measurement errors are assumed to be spatially and temporally uncorrelated and classical, while measurements themselves are non-destructive.

What would settle it

Apply the protocol on a device, then compare the extracted state-preparation and measurement error rates against an independent characterization performed with a different method; significant disagreement would show the separation is inaccurate.

Figures

Figures reproduced from arXiv: 2509.19448 by Leigh M. Norris, Lorenza Viola, Muhammad Qasim Khan.

Figure 1
Figure 1. Figure 1: Pictorial representation of the eight experiments required for the separate characterization of SP and M parameters by using the proposed QSPAM protocol. along with the following combinations: P θz=0 z+→z+→z+ + P θz=π z+→z+→z+ = α 2 M(1 − ϵ) + 2αMα z SP(1 + δ) + (1 + δ) 2 (1 + ϵ) (1 + αMα z SP + δ)(1 + ϵ) , (28) P θz=0 z+→z+→z+ + P θz=π z+→z+→z+ = α 2 M(1 − ϵ) − 2αMα z SP(1 + δ) + (1 + δ) 2 (1 + ϵ) (1 − αM… view at source ↗
Figure 2
Figure 2. Figure 2: In this demonstration, we injected an identical SP errors into qubits [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Characterization of SP and M parameters for arbitrarily selected qubits in the ‘ibm brisbane’ device, where the characterization is implemented over 2 14 shots, with no delay in between each execution of a circuit. The x-axis is the amount of injected SP error in radians, the left column shows the estimated M-error parameter αˆM, and the right column is the SP component along z, αˆ z,(i) SP . The error bar… view at source ↗
Figure 3
Figure 3. Figure 3: Characterization of the SPAM parameters assuming non-diagonal M operators on the ‘ibm brisbane’ device. The experiments were conducted on a set of qubits that are disconnected and spread across the device, with the parameters estimated over 2 15 shots. The results demonstrate a level of SP errors in qubits across the 127-qubit device that is not negligible. The last row shows the contribution of the off-di… view at source ↗
Figure 4
Figure 4. Figure 4: Results for the preparation of a GHZ state with various mitigation strategies applied to the fiducial state provided by a simulated clone of the ‘ibm brisbane’ device with simulated SP and M errors. The estimate of the observable expectation values is constructed over 2 14 shots. The plot on the left depicts a setting where the SP error on each qubit is close to 1%, whereas the plot on the right has a 3.5%… view at source ↗
Figure 5
Figure 5. Figure 5: Preparation of a GHZ state and subsequent estimation of the observable ⟨Z ⊗N ⟩ρGHZ , with various mitigation strategies applied to the fiducial state provided by the ‘ibm sherbrooke’ device. The estimate of the observable expectation values is constructed over 2 14 shots on a linearly connected set of up to 12 qubits. The magenta line shows the result of standard M error mitigation, where SP is assumed to … view at source ↗
read the original abstract

In many platforms, errors from state-preparation and measurement (SPAM) dominate single-qubit gate errors. To inform further hardware improvements and the development of more effective SPAM mitigation strategies, it is necessary to separately characterize the error contributions from state-preparation (SP) and measurement (M). Here, we show how to construct a protocol that can efficiently and separately characterize the SP and M error parameters by using only high-fidelity single-qubit gates and repeated single-qubit measurements without reset. The measurements are assumed to be non-destructive and M errors are taken to be (spatially and temporally) uncorrelated and classical. Notably, the circuit depth of the protocol is independent of system size, and the target parameters may be characterized to a precision that is only limited by the number of experimental repetitions. We employ our protocol for the parallel characterization of SPAM errors on multiple qubits in the IBM Quantum Platform devices, where we find SP infidelities up to $6.57\%$ and readout assignment errors up to $19.1\%$. Using numerical simulations, we also demonstrate how measurement-error mitigation that does not properly account for SP errors generally leads to a biased estimate of measured observable expectation values.

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 / 2 minor

Summary. The manuscript proposes a protocol to separately characterize state-preparation (SP) infidelity and measurement (M) assignment errors using repeated single-qubit measurements without reset and high-fidelity single-qubit gates. Linear combinations of outcome statistics from sequences of varying lengths isolate the two error sources under the assumptions that measurements are non-destructive and that M errors are classical, spatially independent, and temporally uncorrelated. The protocol depth is independent of qubit number. Experimental application on IBM Quantum devices yields SP infidelities up to 6.57% and readout assignment errors up to 19.1%; numerical simulations illustrate bias in observable estimates when SP errors are neglected in mitigation.

Significance. If the stated assumptions hold, the work supplies a scalable, low-depth method for disentangling SP and M contributions that addresses a practical bottleneck in near-term devices. The size-independent circuit depth and parallel multi-qubit characterization are clear strengths, as is the explicit demonstration that ignoring SP errors biases mitigated expectation values. Experimental numbers on real hardware add relevance, though the absence of full data tables, error-propagation details, and robustness checks limits immediate adoption.

major comments (2)
  1. The separation procedure (protocol derivation and linear-combination construction): the closed set of equations used to extract independent SP and M parameters is derived under the strict assumptions of classical, uncorrelated M errors and non-destructive measurements. In superconducting platforms, readout-induced relaxation or weak back-action can introduce temporal correlations or non-classical components that mix the extracted values; the manuscript provides no quantitative bounds or sensitivity analysis on the resulting bias.
  2. Experimental results section: the reported peak values (SP infidelity 6.57%, readout error 19.1%) are stated without accompanying statistical uncertainties, full shot counts per sequence, or explicit propagation of finite-sampling error through the linear inversion. This omission prevents independent verification of the claimed precision limit set only by the number of repetitions.
minor comments (2)
  1. Notation for the assignment probabilities and SP error parameters should be introduced with a clear table or diagram early in the protocol section to aid readability.
  2. The simulation section would benefit from an explicit statement of the noise model used for the 'biased estimate' demonstration, including whether SP errors were modeled as preparation infidelity or as a separate channel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on our manuscript. We address each major comment point by point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: The separation procedure (protocol derivation and linear-combination construction): the closed set of equations used to extract independent SP and M parameters is derived under the strict assumptions of classical, uncorrelated M errors and non-destructive measurements. In superconducting platforms, readout-induced relaxation or weak back-action can introduce temporal correlations or non-classical components that mix the extracted values; the manuscript provides no quantitative bounds or sensitivity analysis on the resulting bias.

    Authors: The protocol derivation explicitly relies on the assumptions of non-destructive measurements and classical, spatially and temporally uncorrelated measurement errors, as stated in the manuscript. These assumptions enable the isolation of SP and M parameters through linear combinations of measurement outcome statistics from sequences of varying lengths. We acknowledge that real superconducting devices may exhibit readout-induced relaxation or weak back-action, potentially introducing correlations not captured by the model. The manuscript focuses on the protocol under the stated assumptions to highlight its efficiency and size-independent depth. In the revised manuscript, we will add a discussion section addressing the robustness to small deviations from these assumptions, including a qualitative analysis of how such effects could bias the extracted parameters. A full quantitative sensitivity analysis would require device-specific noise modeling beyond the scope of this work and is suggested as future research. revision: partial

  2. Referee: Experimental results section: the reported peak values (SP infidelity 6.57%, readout error 19.1%) are stated without accompanying statistical uncertainties, full shot counts per sequence, or explicit propagation of finite-sampling error through the linear inversion. This omission prevents independent verification of the claimed precision limit set only by the number of repetitions.

    Authors: We agree that providing statistical uncertainties, shot counts, and details on error propagation is essential for reproducibility and independent verification. In the revised manuscript, we will include the number of experimental repetitions (shots) used for each sequence length, report the statistical uncertainties on the extracted SP infidelity and readout assignment error values, and describe the propagation of finite-sampling errors through the linear inversion procedure. This will support the claim that precision is limited only by the number of repetitions. revision: yes

Circularity Check

0 steps flagged

Protocol derives SP/M separation from explicit error model without definitional or fitted-input circularity

full rationale

The paper constructs a protocol using sequences of single-qubit gates and repeated non-destructive measurements to form linear combinations whose statistics isolate state-preparation infidelities from readout assignment probabilities. This inversion is derived directly from the assumed classical, uncorrelated, diagonal measurement-error model and the non-destructive projection property; the extracted parameters are not redefined in terms of the same fitted quantities, nor is any central equation shown to reduce to its inputs by construction. No load-bearing self-citations or uniqueness theorems from prior author work are invoked to justify the separation. The derivation remains self-contained once the stated assumptions are granted, with precision limited only by shot statistics rather than by tautological fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The protocol rests on the modeling assumptions stated in the abstract; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Measurements are non-destructive.
    Stated explicitly in the abstract as a prerequisite for repeated single-qubit measurements without reset.
  • domain assumption M errors are spatially and temporally uncorrelated and classical.
    Explicit modeling choice required for the separation of SP and M parameters.

pith-pipeline@v0.9.0 · 5748 in / 1283 out tokens · 29943 ms · 2026-05-18T14:17:06.200892+00:00 · methodology

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 1 internal anchor

  1. [1]

    Degen C L, Reinhard F and Cappellaro P 2017 Quantum sensingRev. Mod. Phys.89035002

    work page 2017

  2. [2]

    Beaudoin F, Norris L M and Viola L 2018 Ramsey interferometry in correlated quantum noise environments Phys. Rev. A98020102

    work page 2018

  3. [3]

    Riberi F, Paz-Silva G A and Viola L 2023 Nearly Heisenberg-limited noise-unbiased frequency estimation by tailored sensor designPhys. Rev. A108042419

    work page 2023

  4. [4]

    Riberi F and Viola L 2025 Optimal asymptotic precision bounds for nonlinear quantum metrology under collective dephasingAPL Quantum2026111

    work page 2025

  5. [5]

    Datta A, Zhang L, Thomas-Peter Net al2011 Quantum metrology with imperfect states and detectorsPhys. Rev. A83063836

    work page

  6. [6]

    136971 Separate characterization of SPAM errors using single-qubit operations21

    Len Y L, Gefen T, Retzker Aet al2022 Quantum metrology with imperfect measurementsNature Commun. 136971 Separate characterization of SPAM errors using single-qubit operations21

    work page

  7. [7]

    Phys.: Cond

    Sza ´nkowski P, Ramon G, Krzywda Jet al2017 Environmental noise spectroscopy with qubits subjected to dynamical decouplingJ. Phys.: Cond. Matter29333001

    work page

  8. [8]

    Paz-Silva G A, Norris L M and Viola L 2017 Multiqubit spectroscopy of Gaussian quantum noisePhys. Rev. A 95022121

    work page 2017

  9. [9]

    Chalermpusitarak T, Tonekaboni B, Wang Yet al2021 Frame-based filter-function formalism for quantum characterization and controlPhys. Rev. X Quantum2030315

    work page

  10. [10]

    Khan M Q, Dong W, Norris L Met al2024 Multiaxis quantum noise spectroscopy robust to errors in state preparation and measurementPhys. Rev. Appl.22024074

    work page

  11. [11]

    Skinner B, Ruhman J and Nahum A 2019 Measurement-induced phase transitions in the dynamics of entanglementPhys. Rev. X9031009

    work page 2019

  12. [12]

    Garratt S J, Weinstein Z and Altman E 2023 Measurements conspire nonlocally to restructure critical quantum statesPhys. Rev. X13021026

    work page 2023

  13. [13]

    Murciano S, Sala P, Liu Yet al2023 Measurement-altered Ising quantum criticalityPhys. Rev. X13041042

    work page

  14. [14]

    Gullans M J and Huse D A 2020 Dynamical purification phase transition induced by quantum measurements Phys. Rev. X10041020

    work page 2020

  15. [15]

    Blunt N S, Caune L, Izs ´ak Ret al2023 Statistical phase estimation and error mitigation on a superconducting quantum processorPhys. Rev. X Quantum4040341

    work page

  16. [16]

    Kang H, Kam J F, Mooney G Jet al2025 Teleporting two-qubit entanglement across 19 qubits on a superconducting quantum computerPhys. Rev. Appl.23014057

    work page

  17. [17]

    Commun.132307

    Livingston W P, Blok M S, Flurin Eet al2022 Experimental demonstration of continuous quantum error correctionNat. Commun.132307

    work page

  18. [18]

    Acharya R, Abanin D A, Aghababaie-Beni Let al2025 Quantum error correction below the surface code thresholdNature638920

    work page

  19. [19]

    An F A, Ransford A, Schaffer Aet al2022 High fidelity state preparation and measurement of ion hyperfine qubits withI > 1 2 Phys. Rev. Lett.129130501

    work page

  20. [20]

    Lin J, Wallman J J, Hincks Iet al2021 Independent state and measurement characterization for quantum computersPhys. Rev. Res.3033285

    work page

  21. [21]

    Bengtsson A, Opremcak A, Khezri M, Sank D, Bourassa A, Satzinger K J, Hong S, Erickson C, Lester B J, Miao K C, Korotkov A N, Kelly J, Chen Z and Klimov P V 2024 Model-based optimization of superconducting qubit readoutPhys. Rev. Lett.132100603

    work page 2024

  22. [22]

    Blumoff J Z, Pan A S, Keating T Eet al2022 Fast and high-fidelity state preparation and measurement in triple-quantum-dot spin qubitsPhys. Rev. X Quantum3010352

    work page

  23. [23]

    Takeda K, Noiri A, Nakajima Tet al2024 Rapid single-shot parity spin readout in a silicon double quantum dot with fidelity exceeding 99%npj Quantum Information1022

    work page

  24. [24]

    Scholl P, Shaw A L, Tsai R B S, Finkelstein R, Choi J and Endres M 2023 Erasure conversion in a high-fidelity Rydberg quantum simulatorNature622273–

    work page 2023

  25. [25]

    Commun.152675

    Chen T, Huang C, Velkovsky I, Hazzard K R A, Covey J P and Gadway B 2024 Strongly interacting Rydberg atoms in synthetic dimensions with a magnetic fluxNat. Commun.152675

    work page 2024

  26. [26]

    Hashim A 2024 A practical introduction to benchmarking and characterization of quantum computers arXiv:2408.12064 [quant-ph]

    work page arXiv 2024

  27. [27]

    Brieger R, Roth I and Kliesch M 2023 Compressive gate set tomographyPhys. Rev. X Quantum4010325

    work page 2023

  28. [28]

    Tech.9035027

    Li Z, Zheng C, Meng Fet al2024 Non-Markovian quantum gate set tomographyQuantum Sci. Tech.9035027

    work page

  29. [29]

    Commun.116301

    White G A L, Hill C D, Pollock F Aet al2020 Demonstration of non-Markovian process characterisation and control on a quantum processorNat. Commun.116301

    work page

  30. [30]

    Hurant T, Sun K, Jia Zet al2024 Few-shot, robust calibration of single qubit gates using Bayesian robust phase estimation (Preprinthttp://arxiv.org/abs/2407.18339:2407.18339)

    work page arXiv

  31. [31]

    Su R Y , Huang J Y , Stuyck N Det al2025 Characterizing non-Markovian quantum processes by fast Bayesian tomographyPhys. Rev. A111052425

    work page

  32. [32]

    Cai Z, Babbush R, Benjamin S C, Endo S, Huggins W J, Li Y , McClean J R and O’Brien T E 2023 Quantum error mitigationRev. Mod. Phys.95045005

    work page 2023

  33. [33]

    Gupta R S, van den Berg E, Takita Met al2024 Probabilistic error cancellation for dynamic quantum circuits Phys. Rev. A109062617

    work page

  34. [34]

    Phys.191116

    van den Berg E, Minev Z K, Kandala Aet al2023 Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processorsNat. Phys.191116

    work page

  35. [35]

    Bravyi S, Sheldon S, Kandala Aet al2021 Mitigating measurement errors in multiqubit experimentsPhys. Rev. A103042605

    work page

  36. [36]

    Nation P D, Kang H, Sundaresan Net al2021 Scalable mitigation of measurement errors on quantum computers Phys. Rev. X Quantum2040326

    work page

  37. [37]

    Ivashkov P, Uchehara G, Jiang Let al2024 High-fidelity, multiqubit generalized measurements with dynamic circuitsPhys. Rev. X Quantum5030315

    work page

  38. [38]

    Yu H and Wei T 2025 Efficient separate quantification of state preparation errors and measurement errors on Separate characterization of SPAM errors using single-qubit operations22 quantum computers and their mitigationQuantum91724

    work page 2025

  39. [39]

    Landa H, Meirom D, Kanazawa Net al2022 Experimental Bayesian estimation of quantum state preparation, measurement, and gate errors in multiqubit devicesPhys. Rev. Res.4013199

    work page

  40. [40]

    Javadi-Abhari A, Treinish M, Krsulich Ket al2024 Quantum computing with QiskitarXiv:2405.08810 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    Maciejewski F B, Zimbor ´as Z and Oszmaniec M 2020 Mitigation of readout noise in near-term quantum devices by classical post-processing based on detector tomographyQuantum4257

    work page 2020

  42. [42]

    Weisberg S 2014 Applied linear regression (John Wiley & Sons, Inc.) chap Appendix A.11, p 309 4th ed maximum Likelihood Estimates

    work page 2014

  43. [43]

    Nielsen E, Gamble J K, Rudinger Ket al2021 Gate Set TomographyQuantum5557

    work page

  44. [44]

    Laflamme R, Lin J and Mor T 2022 Algorithmic cooling for resolving state preparation and measurement errors in quantum computingPhys. Rev. A106012439

    work page 2022

  45. [45]

    Tech.6025009

    Geller M R and Sun M 2021 Toward efficient correction of multiqubit measurement errors: pair correlation methodQuantum Sci. Tech.6025009

    work page 2021

  46. [46]

    Arrasmith A, Patterson A, Boughton Aet al2023 Development and demonstration of an efficient readout error mitigation technique for use in NISQ algorithmsarXiv:2303.17741 [quant-ph]

    work page arXiv

  47. [47]

    Hashim A, Carignan-Dugas A, Chen Let al2025 Quasiprobabilistic readout correction of midcircuit measurements for adaptive feedback via measurement randomized compilingPhys. Rev. X Quantum6 010307

    work page

  48. [48]

    Chen Y , Farahzad M, Yoo Set al2019 Detector tomography on ibm quantum computers and mitigation of an imperfect measurementPhys. Rev. A100052315

    work page