arxiv: 2605.00626 · v1 · submitted 2026-05-01 · 🪐 quant-ph

Recognition: unknown

Learning Lindblad Dynamics of a Superconducting Quantum Processor

Johann Bock Severin , Malthe A. Marciniak , Rune Thinggaard Birke , Emil Hogedal , Andreas Nylander , Irshad Ahmad , Amr Osman , Janka Bizn\'arov\'a

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Marcus Rommel Anita Fadavi Roudsari Jonas Bylander Giovanna Tancredi Christopher W. Warren Svend Kr{\o}jer Jacob Hastrup Morten Kjaergaard

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:49 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Lindblad modelsquantum processor characterizationmodel selectionsuperconducting qubitstomographic datalikelihood ratio testsnoise modelingmulti-qubit interactions

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The pith

A statistical framework learns minimal Lindblad models for quantum processors by testing which terms the data actually require.

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

The paper introduces LIMINAL, a method that builds compact models of quantum processor dynamics without assuming in advance which interactions or noise processes matter. It fits a sequence of nested Lindblad models to time-resolved tomographic measurements and applies likelihood-ratio tests to retain only those Hamiltonian or dissipation terms that measurably improve the description. On a five-qubit superconducting device the procedure selects three-local Hamiltonian terms plus two-local dissipation for idling while rejecting three-local dissipation. The same workflow recovers driven single-qubit Hamiltonians, reconstructs shaped-pulse evolution, and checks for hidden-qubit effects in coupler dynamics. The result matters because models that are either too simple or unnecessarily elaborate both limit the ability to predict and improve device performance.

Core claim

LIMINAL fits nested candidate Lindblad models to time-resolved tomographic data and uses likelihood-ratio tests to decide when added physical mechanisms are warranted, producing an idling model for a five-qubit superconducting processor that includes three-local Hamiltonian terms and two-local dissipation but finds no support for three-local dissipation.

What carries the argument

LIMINAL, the framework that performs nested model fitting to tomographic data followed by likelihood-ratio tests to validate inclusion of specific multi-local Hamiltonian or dissipation terms.

Load-bearing premise

The time-resolved tomographic data are sufficiently informative and free of systematic errors so that the likelihood-ratio tests can reliably distinguish which mechanisms are present.

What would settle it

Apply the likelihood-ratio test to new tomographic data recorded after deliberately engineering a three-local dissipation channel on the same five-qubit processor and verify whether the test now supports adding that term.

Figures

Figures reproduced from arXiv: 2605.00626 by Amr Osman, Andreas Nylander, Anita Fadavi Roudsari, Christopher W. Warren, Emil Hogedal, Giovanna Tancredi, Irshad Ahmad, Jacob Hastrup, Janka Bizn\'arov\'a, Johann Bock Severin, Jonas Bylander, Malthe A. Marciniak, Marcus Rommel, Morten Kjaergaard, Rune Thinggaard Birke, Svend Kr{\o}jer.

**Figure 1.** Figure 1: FIG. 1. Conceptual overview of the framework for learning view at source ↗

**Figure 2.** Figure 2: FIG. 2. Application of the LIMINAL framework to determine the minimal adequate model for the idling dynamics of a 5-qubit view at source ↗

**Figure 3.** Figure 3: FIG. 3. Single-qubit validation under a constant, slightly de view at source ↗

**Figure 4.** Figure 4: FIG. 4. Reconstruction of a time-dependent Hamiltonian view at source ↗

**Figure 5.** Figure 5: FIG. 5. Capturing coupler-mediated entangling dynamics and non-Markovian memory with minimal hidden qubit models. view at source ↗

**Figure 6.** Figure 6: FIG. 6 view at source ↗

**Figure 7.** Figure 7: FIG. 7. IQ data showing the distribution of single shot data for the five different qubits. view at source ↗

**Figure 8.** Figure 8: FIG. 8. Fidelities for different operations in the gate set extracted via interleaved randomized benchmarking [52]. The right view at source ↗

**Figure 9.** Figure 9: FIG. 9. Fidelities for identities constructed from the view at source ↗

**Figure 10.** Figure 10: FIG. 10. Random selection of 18 configurations out of 248,832 different tomographical experiments. Three models are visualized, view at source ↗

**Figure 11.** Figure 11: FIG. 11 view at source ↗

**Figure 12.** Figure 12: FIG. 12 view at source ↗

**Figure 13.** Figure 13: FIG. 13. Overview of fit and data for all visualization of the enveloped function. The title refers to initialized state and the view at source ↗

**Figure 14.** Figure 14: FIG. 14. Overview of fit and data for all visualization of the 1 qubit tomography of the driven coupler experiment. The title view at source ↗

**Figure 15.** Figure 15: FIG. 15. Overview of fit and data for 12 randomly selected configurations of the 2 qubit tomography of the driven coupler view at source ↗

read the original abstract

Accurate models of quantum processors are essential for understanding, calibrating, and improving their performance. In practice, model construction must balance physical detail against the experimental and computational effort required to reliably learn parameters. Compact descriptions therefore often rely on assumptions about which interactions, noise processes, or hidden degrees of freedom are relevant. Here we introduce LIMINAL, a data-driven framework for testing such assumptions and selecting minimal adequate Lindblad models. LIMINAL fits nested candidate models to time-resolved tomographic data and uses likelihood-ratio tests to decide when added physical mechanisms are warranted. We apply LIMINAL to a five-qubit superconducting processor, identifying an idling model with three-local Hamiltonian terms and two-local dissipation, while finding no support for three-local dissipation. We further apply it to recover driven single-qubit Hamiltonians, reconstruct a shaped-pulse Hamiltonian without assuming an analytic pulse model, and test hidden-qubit extensions in coupler-mediated dynamics, demonstrating the applicability of the framework for a wide range of tasks.

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.

Desk Editor's Note private letter to a colleague

LIMINAL gives a practical likelihood-ratio procedure for picking minimal Lindblad models from tomographic data on superconducting qubits, with the headline claim that three-local Hamiltonian terms are needed but three-local dissipation is not.

read the letter

LIMINAL fits nested Lindblad models to time-resolved tomographic data and uses likelihood-ratio tests to decide which extra terms are justified by the data. On a five-qubit superconducting processor they report that an idling model needs three-local Hamiltonian terms plus two-local dissipation, but finds no evidence for three-local dissipation. They also show the method on driven single-qubit cases, pulse reconstruction without an assumed shape, and hidden-qubit tests in coupler dynamics. That combination of nested fitting plus formal model selection on real hardware data is the main new piece. The approach is straightforward and avoids fixing the model form by hand, which is useful when the relevant interactions are not obvious in advance. Applying it across idling, driven, and coupler settings shows the framework is not limited to one narrow task. The experimental data come from an actual device, so the results are grounded in hardware rather than simulation alone. A clear soft spot is the dependence on the likelihood-ratio test behaving as expected. The test assumes the data are generated by one of the candidate models with no unmodeled systematics. Superconducting tomography is known to suffer from readout drifts and incomplete basis coverage, and if those effects are comparable to the weak three-local signals they can produce spurious likelihood gains or hide real ones. The abstract gives no fit statistics, p-values, or checks against synthetic data with added errors, so it is hard to judge how robust the inclusion decisions actually are. Readers working on quantum processor characterization and calibration will find the method and the specific five-qubit findings directly relevant. Experimental groups scaling superconducting hardware or similar platforms get the most value; theorists focused on open-system modeling may also pick up the selection procedure. The paper deserves peer review because the framework is usable and the hardware demonstration is concrete, even if the statistical assumptions need closer scrutiny.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces LIMINAL, a framework that fits nested candidate Lindblad models to time-resolved tomographic data and applies likelihood-ratio tests to select minimal adequate models by determining which physical mechanisms (Hamiltonian terms, dissipation channels) are statistically supported. Applied to a five-qubit superconducting processor, it reports an idling model requiring three-local Hamiltonian terms and two-local dissipation while finding no support for three-local dissipation; additional demonstrations include recovery of driven single-qubit Hamiltonians, reconstruction of a shaped-pulse Hamiltonian, and tests of hidden-qubit extensions in coupler-mediated dynamics.

Significance. If the statistical tests prove robust, LIMINAL could provide a principled, data-driven alternative to ad-hoc model assumptions in quantum processor characterization, enabling more compact yet accurate Lindblad descriptions that improve calibration, simulation, and error mitigation. The use of likelihood-ratio tests on tomographic time series is a methodological strength when data are high-quality and free of unmodeled systematics.

major comments (3)

[five-qubit idling experiment] In the five-qubit idling experiment: the headline result that three-local dissipation is excluded while three-local Hamiltonian terms are required depends on the likelihood-ratio tests correctly attributing structure to the data. No test statistics, p-values, degrees of freedom, or effective data volume are reported, so it is impossible to judge test power or whether weak three-local signals could be masked or spuriously generated.
[LIMINAL framework] In the LIMINAL framework description: the method assumes tomographic data are generated exactly by some Lindblad model within the candidate set so that the test statistic follows the asymptotic chi-squared distribution. Time-resolved tomography on superconducting hardware is susceptible to readout errors, calibration drift, and incomplete Pauli coverage; any such systematics at the level of the reported weak three-local signals would invalidate the inclusion/exclusion decisions.
[shaped-pulse application] In the shaped-pulse Hamiltonian reconstruction: the claim of recovering the Hamiltonian without assuming an analytic pulse model requires explicit parameterization details and a comparison of model complexity (number of free parameters) against a baseline analytic model to confirm that the likelihood improvement is not simply due to added flexibility.

minor comments (2)

[abstract] The abstract would be strengthened by a brief statement of data volume (number of time points or total measurements) and at least one quantitative fit metric to support the model-selection claims.
[methods] Notation for the candidate model sets and the exact form of the likelihood function should be defined consistently in the methods section before the experimental applications.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment point by point below, indicating the revisions made.

read point-by-point responses

Referee: [five-qubit idling experiment] In the five-qubit idling experiment: the headline result that three-local dissipation is excluded while three-local Hamiltonian terms are required depends on the likelihood-ratio tests correctly attributing structure to the data. No test statistics, p-values, degrees of freedom, or effective data volume are reported, so it is impossible to judge test power or whether weak three-local signals could be masked or spuriously generated.

Authors: We agree that the test statistics are necessary to allow readers to assess the strength of evidence and test power. In the revised manuscript we have added Table II, which reports the likelihood-ratio test statistics, p-values, degrees of freedom, and effective data volume for all nested-model comparisons involving three-local Hamiltonian and dissipation terms in the five-qubit idling data set. These values confirm that the inclusion of three-local Hamiltonian terms is statistically supported while three-local dissipation is not, at the reported significance level. revision: yes
Referee: [LIMINAL framework] In the LIMINAL framework description: the method assumes tomographic data are generated exactly by some Lindblad model within the candidate set so that the test statistic follows the asymptotic chi-squared distribution. Time-resolved tomography on superconducting hardware is susceptible to readout errors, calibration drift, and incomplete Pauli coverage; any such systematics at the level of the reported weak three-local signals would invalidate the inclusion/exclusion decisions.

Authors: The referee correctly notes the central modeling assumption. We have expanded Section II.C to state this assumption explicitly and to discuss its sensitivity to unmodeled systematics. We have also added a new paragraph quantifying the magnitude of readout or calibration errors that would be required to alter the reported model-selection outcomes. While a full experimental characterization of every possible systematic is outside the scope of the present work, the added discussion clarifies the conditions under which the likelihood-ratio decisions remain valid. revision: partial
Referee: [shaped-pulse application] In the shaped-pulse Hamiltonian reconstruction: the claim of recovering the Hamiltonian without assuming an analytic pulse model requires explicit parameterization details and a comparison of model complexity (number of free parameters) against a baseline analytic model to confirm that the likelihood improvement is not simply due to added flexibility.

Authors: We have added the requested parameterization details to the main text (Section IV.C) and to the supplementary material, specifying the time-dependent basis functions and the total number of free parameters (42 for the flexible model versus 12 for the analytic baseline). We further report the likelihood-ratio statistic comparing the two models, demonstrating that the improvement remains significant after accounting for the difference in degrees of freedom. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework applies standard fitting and tests to external data

full rationale

The derivation chain consists of fitting nested Lindblad models to time-resolved tomographic data from a superconducting processor and applying likelihood-ratio tests for model selection. These steps rely on external experimental inputs and standard statistical procedures rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The reported idling model (three-local Hamiltonian terms present, three-local dissipation absent) is a direct output of the data-driven tests, not equivalent to the inputs by construction. The paper remains self-contained against external benchmarks with no reduction of claims to tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard Lindblad form being an adequate description and on the statistical power of likelihood-ratio tests applied to tomographic data. No free parameters are mentioned. The main addition is the LIMINAL procedure itself.

axioms (1)

domain assumption Quantum processor dynamics can be adequately described by a Lindblad master equation.
Invoked throughout the framework description as the model class being selected.

invented entities (1)

LIMINAL framework no independent evidence
purpose: Data-driven selection of minimal Lindblad models via nested fitting and likelihood-ratio tests.
New procedure introduced in the paper; no independent evidence outside this work.

pith-pipeline@v0.9.0 · 5542 in / 1252 out tokens · 44015 ms · 2026-05-09T19:49:29.462048+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

54 extracted references · 14 canonical work pages · 2 internal anchors

[1]

Siddiqi, Engineering high-coherence superconducting qubits, Nature Reviews Materials6, 875 (2021)

I. Siddiqi, Engineering high-coherence superconducting qubits, Nature Reviews Materials6, 875 (2021)

2021
[2]

N. Wittler, Integrated Tool Set for Control, Calibration, and Characterization of Quantum Devices Applied to Superconducting Qubits, Physical Review Applied15, 10.1103/PhysRevApplied.15.034080 (2021)

work page doi:10.1103/physrevapplied.15.034080 2021
[3]

H. R. Naeij, Open quantum system approaches to super- conducting qubits, Quantum Information Processing24, 220 (2025)

2025
[4]

Hashim, L

A. Hashim, L. B. Nguyen, N. Goss, B. Marinelli, R. K. Naik, T. Chistolini, J. Hines, J. P. Marceaux, Y. Kim, P. Gokhale, T. Tomesh, S. Chen, L. Jiang, S. Ferracin, K. Rudinger, T. Proctor, K. C. Young, I. Siddiqi, and R. Blume-Kohout, A Practical Introduction to Bench- marking and Characterization of Quantum Comput- ers, PRX Quantum6, 030202 (2025), arXiv...

work page arXiv 2025
[5]

Brieger, Compressive Gate Set Tomography, PRX Quantum4, 10.1103/PRXQuantum.4.010325 (2023)

R. Brieger, Compressive Gate Set Tomography, PRX Quantum4, 10.1103/PRXQuantum.4.010325 (2023)

work page doi:10.1103/prxquantum.4.010325 2023
[6]

Viñas and A

P. Viñas and A. Bermudez, Microscopic parametrizations for gate set tomography under coloured noise, npj Quan- tum Information11, 23 (2025)

2025
[7]

J.F.Poyatos, J.I.Cirac,andP.Zoller,CompleteCharac- terization of a Quantum Process: The Two-Bit Quantum Gate, Physical Review Letters78, 390 (1997)

1997
[8]

G. C. Knee, Quantum process tomography via com- pletely positive and trace-preserving projection, Physical Review A98, 10.1103/PhysRevA.98.062336 (2018)

work page doi:10.1103/physreva.98.062336 2018
[9]

G. O. Samach, A. Greene, J. Borregaard, M. Christandl, J. Barreto, D. K. Kim, C. M. McNally, A. Melville, B. M. Niedzielski, Y. Sung, D. Rosenberg, M. E. Schwartz, J. L. Yoder, T. P. Orlando, J. I.-J. Wang, S. Gustavsson, M. Kjaergaard, and W. D. Oliver, Lindblad Tomogra- phy of a Superconducting Quantum Processor, Physical Review Applied18, 064056 (2022)

2022
[10]

Varona, M

S. Varona, M. Müller, and A. Bermudez, Lindblad-like quantum tomography for non-Markovian quantum dy- namical maps, npj Quantum Information11, 96 (2025)

2025
[11]

Wallace, Learning the dynamics of Markovian open quantum systems from experimental data, Physical Re- view Research7, 10.1103/m66m-lnjh (2025)

S. Wallace, Learning the dynamics of Markovian open quantum systems from experimental data, Physical Re- view Research7, 10.1103/m66m-lnjh (2025)

work page doi:10.1103/m66m-lnjh 2025
[12]

Cramer, M

M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient quantum state tomography, Na- ture Communications1, 149 (2010)

2010
[13]

Nielsen, J

E. Nielsen, J. K. Gamble, K. Rudinger, T. Scholten, K. Young, and R. Blume-Kohout, Gate Set Tomography, Quantum5, 557 (2021)

2021
[14]

Stilck França, L

D. Stilck França, L. A. Markovich, V. V. Dobrovitski, A. H. Werner, and J. Borregaard, Efficient and robust estimation of many-qubit Hamiltonians, Nature Commu- nications15, 311 (2024)

2024
[15]

Hangleiter, I

D. Hangleiter, I. Roth, J. Fuksa, J. Eisert, and P. Roushan, Robustly learning the Hamiltonian dynam- ics of a superconducting quantum processor, Nature Communications15, 9595 (2024)

2024
[16]

S. Liu, X. Wu, and M. Y. Niu, Optimal and Robust In- situ Quantum Hamiltonian Learning through Paralleliza- tion (2025), arXiv:2510.07818 [quant-ph]

work page arXiv 2025
[17]

R. T. Birke, J. B. Severin, M. A. Marciniak, E. Hogedal, A. Nylander, I. Ahmad, A. Osman, J. Biznárová, M. Rommel, A. F. Roudsari, J. Bylander, G. Tancredi, D. S. França, A. Werner, C. W. Warren, J. Hastrup, S.Krøjer,andM.Kjaergaard,DemonstratingandBench- marking Classical Shadows for Lindblad Tomography (2026), arXiv:2602.14694 [quant-ph]

work page arXiv 2026
[18]

S. S. Wilks, The Large-Sample Distribution of the Likeli- hood Ratio for Testing Composite Hypotheses, The An- nals of Mathematical Statistics9, 60 (1938)

1938
[19]

Bradbury, R

J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. Vander- Plas, S. Wanderman-Milne, and Q. Zhang, JAX: Com- posable transformations of Python+NumPy programs (2018)

2018
[20]

(2022, 2)

P. Kidger, On Neural Differential Equations (2022), arXiv:2202.02435 [cs]

work page arXiv 2022
[21]

Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976)

1976
[22]

Paris and J

M. Paris and J. Rehacek,Quantum State Estimation (Springer Science & Business Media, 2004)

2004
[23]

A. W. Harrow, A. Montanaro, and A. J. Short, Limita- tionsonquantumdimensionalityreduction,International Journal of Quantum Information13, 1440001 (2015)

2015
[24]

BRADLEY.EFRONandD.V.HINKLEY,Assessingthe accuracy of the maximum likelihood estimator: Observed versus expected Fisher information, Biometrika65, 457 (1978)

1978
[25]

Akaike, A new look at the statistical model identifica- tion, IEEE Transactions on Automatic Control19, 716 (1974)

H. Akaike, A new look at the statistical model identifica- tion, IEEE Transactions on Automatic Control19, 716 (1974)

1974
[26]

Schwarz, Estimating the Dimension of a Model, The Annals of Statistics6, 461 (1978), 2958889

G. Schwarz, Estimating the Dimension of a Model, The Annals of Statistics6, 461 (1978), 2958889

1978
[27]

Bairey, I

E. Bairey, I. Arad, and N. H. Lindner, Learning a Local Hamiltonian from Local Measurements, Physical Review Letters122, 020504 (2019)

2019
[28]

J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric Informationally Complete Quantum Measurements, Journal of Mathematical Physics45, 2171 (2004), arXiv:quant-ph/0310075

work page arXiv 2004
[29]

Krantz, M

P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gus- tavsson, and W. D. Oliver, A Quantum Engineer’s Guide to Superconducting Qubits, Applied Physics Reviews6, 021318 (2019), arXiv:1904.06560 [quant-ph]

work page arXiv 2019
[30]

Chiorescu, Y

I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij, Coherent Quantum Dynamics of a Super- conducting Flux Qubit, Science299, 1869 (2003)

2003
[31]

Motzoi, J

F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Simple Pulses for Elimination of Leakage in Weakly Nonlinear Qubits, Physical Review Letters103, 110501 (2009)

2009
[32]

Hyyppä, A

E. Hyyppä, A. Vepsäläinen, M. Papič, C. F. Chan, S. Inel, A. Landra, W. Liu, J. Luus, F. Marxer, C. Ockeloen-Korppi, S. Orbell, B. Tarasinski, and J. Heinsoo, Reducing Leakage of Single-Qubit Gates for Superconducting Quantum Processors Using Analyt- ical Control Pulse Envelopes, PRX Quantum5, 030353 (2024)

2024
[33]

J. M. Gambetta, F. Motzoi, S. T. Merkel, and F. K. Wil- helm, Analytic control methods for high-fidelity unitary operations in a weakly nonlinear oscillator, Physical Re- view A83, 012308 (2011). 14

2011
[34]

Chow, and J

D.C.McKay, S.Filipp, A.Mezzacapo, E.Magesan, J.M. Chow, and J. M. Gambetta, Universal Gate for Fixed- Frequency Qubits via a Tunable Bus, Physical Review Applied6, 064007 (2016)

2016
[35]

Blais, A

A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wall- raff, Circuit quantum electrodynamics, Reviews of Mod- ern Physics93, 025005 (2021)

2021
[36]

M. Odeh, K. Godeneli, E. Li, R. Tangirala, H. Zhou, X. Zhang, Z.-H. Zhang, and A. Sipahigil, Non-Markovian dynamics of a superconducting qubit in a phononic bandgap, Nature Physics21, 406 (2025)

2025
[37]

S.Nakajima,OnQuantumTheoryofTransportPhenom- ena: Steady Diffusion, Progress of Theoretical Physics 20, 948 (1958)

1958
[38]

Zwanzig, Ensemble Method in the Theory of Irre- versibility, Journal of Chemical Physics33, 1338 (1960)

R. Zwanzig, Ensemble Method in the Theory of Irre- versibility, Journal of Chemical Physics33, 1338 (1960)

1960
[39]

Shnirman, Low- and High-Frequency Noise from Co- herent Two-Level Systems, Physical Review Letters94, 10.1103/PhysRevLett.94.127002 (2005)

A. Shnirman, Low- and High-Frequency Noise from Co- herent Two-Level Systems, Physical Review Letters94, 10.1103/PhysRevLett.94.127002 (2005)

work page doi:10.1103/physrevlett.94.127002 2005
[40]

Anto-Sztrikacs and D

N. Anto-Sztrikacs and D. Segal, Capturing non- Markovian dynamics with the reaction coordinate method, Physical Review A104, 052617 (2021)

2021
[41]

Bengtsson, J

C.W.Warren, J.Fernández-Pendás, S.Ahmed, T.Abad, A. Bengtsson, J. Biznárová, K. Debnath, X. Gu, C. Križan, A. Osman, A. Fadavi Roudsari, P. Delsing, G. Johansson, A. Frisk Kockum, G. Tancredi, and J. By- lander, Extensive characterization and implementation of a family of three-qubit gates at the coherence limit, npj Quantum Information9, 44 (2023)

2023
[42]

J. R. Schrieffer and P. A. Wolff, Relation between the An- derson and Kondo Hamiltonians, Physical Review149, 491 (1966)

1966
[43]

M. Roth, M. Ganzhorn, N. Moll, S. Filipp, G. Salis, and S.Schmidt,Analysisofaparametricallydrivenexchange- type gate and a two-photon excitation gate between su- perconducting qubits, Physical Review A96, 062323 (2017)

2017
[44]

Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghe- mawat, Ian Goodfellow, Andrew Harp, Geoffrey Irv- ing, Michael Isard, Y

Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghe- mawat, Ian Goodfellow, Andrew Harp, Geoffrey Irv- ing, Michael Isard, Y. Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dande- lion Mané, Rajat Monga, Sherry Moore, Derek Mur- ...

2015
[45]

Babuschkin, K

DeepMind, I. Babuschkin, K. Baumli, A. Bell, S. Bhu- patiraju, J. Bruce, P. Buchlovsky, D. Budden, T. Cai, A. Clark, I. Danihelka, A. Dedieu, C. Fantacci, J. God- win, C. Jones, R. Hemsley, T. Hennigan, M. Hes- sel, S. Hou, S. Kapturowski, T. Keck, I. Kemaev, M. King, M. Kunesch, L. Martens, H. Merzic, V. Miku- lik, T. Norman, G. Papamakarios, J. Quan, R....

2020
[46]

Tsitouras, Runge–Kutta pairs of order 5 (4) satisfying only the first column simplifying assumption, Computers & Mathematics with Applications62, 770 (2011)

C. Tsitouras, Runge–Kutta pairs of order 5 (4) satisfying only the first column simplifying assumption, Computers & Mathematics with Applications62, 770 (2011)

2011
[47]

Stumm and A

P. Stumm and A. Walther, New Algorithms for Optimal Online Checkpointing, SIAM Journal on Scientific Com- puting32, 836 (2010)

2010
[48]

Q. Wang, P. Moin, and G. Iaccarino, Minimal Repetition Dynamic Checkpointing Algorithm for Unsteady Adjoint Calculation, SIAM Journal on Scientific Computing31, 2549 (2009)

2009
[49]

R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Du- venaud, Neural Ordinary Differential Equations (2019), arXiv:1806.07366 [cs]

work page internal anchor Pith review arXiv 2019
[50]

D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization (2017), arXiv:1412.6980 [cs]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[51]

M. A. Marciniak, R. T. Birke, J. B. Severin, F. Berritta, D. Kjær, F. Nilsson, S. N. Themadath, S. Kallatt, J. L. Webb, K. Bentsen, T. Madsen, Z. Sun, S. Krøjer, C. W. Warren, J. Hastrup, and M. Kjaergaard, Millisecond- Scale Calibration and Benchmarking of Superconducting Qubits (2026), arXiv:2602.11912 [quant-ph]

work page arXiv 2026
[52]

Magesan, J

E. Magesan, J. M. Gambetta, B. R. Johnson, C. A. Ryan, J. M. Chow, S. T. Merkel, M. P. da Silva, G. A. Keefe, M. B. Rothwell, T. A. Ohki, M. B. Ketchen, and M. Stef- fen, Efficient Measurement of Quantum Gate Error by Interleaved Randomized Benchmarking, Physical Review Letters109, 080505 (2012)

2012
[53]

Magesan, J

E. Magesan, J. M. Gambetta, and J. Emerson, Char- acterizing quantum gates via randomized benchmarking, Physical Review A85, 042311 (2012)

2012
[54]

0”| |1⟩)−p(“1

D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficient $Z$ gates for quantum com- puting, Physical Review A96, 022330 (2017). 15 Appendix A: Practical Implementation Details Inthissection, weprovideadditionaldetailsonthepracticalimplementationoftheclassicalanalysis. Allsimulations and optimizations are implemented in Python usingJA...

2017