Which Superconducting Qubit Model is Good Enough? From Effective Two-Level to Circuit-Based Hamiltonians for Pulse-Level Simulation

Frej Larssen; Ivy Peng; Stefano Markidis

arxiv: 2605.23034 · v1 · pith:IPVK43ENnew · submitted 2026-05-21 · 💻 cs.ET

Which Superconducting Qubit Model is Good Enough? From Effective Two-Level to Circuit-Based Hamiltonians for Pulse-Level Simulation

Frej Larssen , Ivy Peng , Stefano Markidis This is my paper

Pith reviewed 2026-05-25 05:15 UTC · model grok-4.3

classification 💻 cs.ET

keywords superconducting qubitsHamiltonian modelspulse-level simulationDuffing modeltransmon circuitflux-tunable qubitleakage analysistwo-qubit gates

0 comments

The pith

The Duffing model follows the circuit-based reference more closely than the effective two-level model for static spectra and reduced two-qubit quantities in a flux-tunable device.

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

This paper compares three Hamiltonian models for pulse-level simulation of the same flux-tunable two-qubit superconducting device with fixed bus coupler: an effective two-level model, a three-mode Duffing model, and a circuit-based transmon model in the charge basis. Using a realistic parameter set, the models are tested on benchmarks that include flux-dependent spectra, extracted two-qubit interaction terms, driven single-qubit dynamics, CZ gate evolution, leakage outside the computational subspace, and runtime. The Duffing model stays closer to the circuit reference than the effective model on static spectra and two-qubit quantities. In driven cases the multilevel models display effects that the effective description misses entirely. The results lead to the recommendation of layered abstraction, with effective models for reduced analyses, Duffing models as a practical multilevel default, and circuit models for high-fidelity reference or leakage studies.

Core claim

Across the tested flux range, the Duffing model follows the circuit-based reference more closely than the effective model for static spectra and reduced two-qubit quantities, while in driven benchmarks, the multilevel models reveal effects absent in the effective description. Overall, the results support a layered use of abstraction in pulse-level simulation: effective models for reduced analyses, Duffing models as a practical multilevel default, and circuit-based models for high-fidelity reference simulation or detailed leakage analysis.

What carries the argument

Benchmark comparison of three Hamiltonian descriptions (effective two-level, three-mode Duffing, circuit-based transmon in charge basis) on a common suite for a flux-tunable two-qubit device with fixed bus coupler.

If this is right

Effective two-level models remain usable only when the simulation goal is limited to reduced analyses of spectra or interaction terms.
Duffing models serve as a practical default for most multilevel pulse-level simulations because they track the circuit reference more closely than effective models.
Circuit-based models are required when the objective demands high-fidelity reference results or detailed leakage analysis.
A layered abstraction strategy lets simulators match model complexity to the specific simulation objective.

Where Pith is reading between the lines

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

If the tested parameter set is typical, many existing pulse simulators could adopt Duffing models by default to gain accuracy at modest extra cost over effective models.
The benchmark differences suggest that future simulators could include automatic model selection that switches between effective, Duffing, and circuit levels according to the required fidelity for spectra versus driven dynamics.
Repeating the same comparison on devices with different coupler designs or larger qubit counts would test how far the layered-abstraction recommendation extends.

Load-bearing premise

The chosen realistic parameter set for the flux-tunable two-qubit device with fixed bus coupler is representative of real hardware, and the benchmark suite is comprehensive enough to justify the general recommendation for layered abstraction in pulse-level simulation.

What would settle it

Direct experimental measurements of flux-dependent spectra, two-qubit interaction strengths, and leakage rates during driven gates on the physical device, compared against predictions from each of the three models, would determine which description matches hardware most closely.

Figures

Figures reproduced from arXiv: 2605.23034 by Frej Larssen, Ivy Peng, Stefano Markidis.

**Figure 1.** Figure 1: Two flux-tunable transmon qubits (𝑞0 on the right and 𝑞1 on the left) are coupled through a fixed harmonic bus, represented as an intermediate LC resonator. The subsystem ordering is |𝑞1, 𝑐, 𝑞0⟩, with 𝑞0 the least significant qubit. Each transmon is coupled to a microwave drive line, modeled here by the time-dependent voltage 𝑉𝑑,𝑗 (𝑡). of the model definition itself. It determines the dimension of the matr… view at source ↗

**Figure 3.** Figure 3: Static comparison across flux on qubit 𝑞1. The panels show dressed computational energies relative to the ground state, per-flux spectral RMSE with respect to the circuit-based model, the extracted exchange-like coupling 𝐽 (𝜙), and the residual interaction 𝜁 (𝜙). 0.0 0.5 1.0 Population Population |00⟩→|01⟩ 0 5 10 15 20 25 Time (ns) 0.0 0.5 1.0 Population Population |10⟩→|11⟩ Circuit Duffing Effective [PIT… view at source ↗

**Figure 4.** Figure 4: 𝑅𝑋 benchmark on 𝑞0 with spectator qubit 𝑞1 initialized in |0⟩ (top) and |1⟩ (bottom). The curves show the population transfer within the computational subspace predicted by the three models under the same drive pulse. the intended computational transition but also additional degrees of freedom associated with higher qubit levels and the shared bus. These differences become clearer in [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 6.** Figure 6: CZ gate dynamics under a flux pulse applied to [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 5.** Figure 5: Diagnostics for the driven 𝑅𝑋 benchmark. Top and middle: leakage for the two spectator initializations. Bottom: spectator-state mismatch, defined as the absolute difference between the population transfer curves for |00⟩ → |01⟩ and |10⟩ → |11⟩. response of the driven qubit depends on the initial state of the spectator qubit, and would vanish for an ideal spectator-independent single-qubit rotation. The ef… view at source ↗

**Figure 7.** Figure 7: Leakage pathways during the CZ pulse for an initial [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: Runtime of the CZ simulation as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗

read the original abstract

Pulse-level simulators are the lowest-level, most widely used abstraction layer for studying how quantum hardware responds to control signals, but they can be built on Hamiltonian models with very different fidelity and cost. This raises the question: which level of physical abstraction is sufficient for a given simulation objective? We study this question for a flux-tunable two-qubit superconducting device with a fixed bus coupler by comparing three Hamiltonian descriptions of the same hardware: an effective two-level model, a three-mode Duffing model, and a circuit-based transmon model in the charge basis. Using a realistic parameter set, we evaluate these models on a common benchmark suite spanning flux-dependent spectra, extracted two-qubit interaction terms, driven single-qubit dynamics, CZ gate dynamics, leakage outside the computational subspace, and runtime. Across the tested flux range, the Duffing model follows the circuit-based reference more closely than the effective model for static spectra and reduced two-qubit quantities, while in driven benchmarks, the multilevel models reveal effects absent in the effective description. Overall, the results support a layered use of abstraction in pulse-level simulation: effective models for reduced analyses, Duffing models as a practical multilevel default, and circuit-based models for high-fidelity reference simulation or detailed leakage analysis.

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

The paper runs a head-to-head of effective two-level, Duffing, and circuit models on one flux-tunable two-qubit device and finds Duffing tracks the circuit reference better on static quantities while multilevel models catch driven leakage the effective model misses.

read the letter

The central result is that, for the chosen realistic parameters, the Duffing model stays closer to the circuit-based reference than the effective two-level model does on flux-dependent spectra and reduced two-qubit interaction terms. In driven cases the multilevel descriptions also surface leakage and other effects that the effective model simply does not produce. The authors therefore suggest a practical layering: effective models for quick reduced analyses, Duffing as the default multilevel choice, and circuit models only when high-fidelity leakage checks are needed. They also report runtimes for the three approaches on the same benchmark suite covering spectra, CZ dynamics, and single-qubit drive. That side-by-side evaluation on this device type is the concrete new piece; prior work has compared subsets of these models but not all three together on exactly these observables for a flux-tunable pair with fixed bus coupler. The comparison itself is straightforward and the benchmarks are standard for the subfield, so the numbers on closeness and cost are usable as a reference point. The main limitation is that every quantitative claim rests on a single fixed parameter set. There are no sweeps over anharmonicity, coupling strength, or flux range beyond the tested interval, and no cross-device checks. This makes the general recommendation for layered abstraction harder to apply outside the specific values chosen. The paper is written for engineers and simulator developers who need to pick a Hamiltonian level for pulse-level work on superconducting hardware. A reader already running similar simulations will get immediate practical guidance from the numbers. It is coherent on its own terms and engages the existing literature on model fidelity, so it is worth sending to peer review even though the scope is narrow and the single-parameter design will probably draw reviewer comments on generalizability.

Referee Report

1 major / 2 minor

Summary. The manuscript compares three Hamiltonian models for a flux-tunable two-qubit superconducting device with fixed bus coupler: an effective two-level model, a three-mode Duffing model, and a circuit-based transmon model in the charge basis. Using one realistic parameter set, the models are benchmarked on flux-dependent spectra, extracted two-qubit interaction terms, driven single-qubit and CZ dynamics, leakage, and runtime. The central claim is that the Duffing model tracks the circuit reference more closely than the effective model for static spectra and reduced two-qubit quantities, while multilevel models capture driven effects absent from the effective description, supporting a layered abstraction strategy (effective for reduced analyses, Duffing as practical default, circuit for high-fidelity reference).

Significance. If the reported closeness metrics and differential driven effects hold under the chosen parameters, the work supplies concrete, actionable guidance on Hamiltonian abstraction levels for pulse-level simulators, with the multi-aspect benchmark suite (spectra through leakage and runtime) serving as a reusable evaluation template. This addresses a practical question in quantum control simulation with direct relevance to hardware modeling workflows.

major comments (1)

[Abstract] Abstract: the general recommendation for layered abstraction in pulse-level simulation rests on comparisons performed with a single fixed 'realistic parameter set' for one flux-tunable two-qubit device; no sweeps over anharmonicity, coupling strength, or flux range beyond the tested interval, nor cross-device validation, are reported, so the quantitative superiority of the Duffing model and the absence of effective-model effects in driven cases may be specific to the chosen values rather than generic.

minor comments (2)

The abstract states comparative outcomes but does not reference specific figures, tables, or quantitative metrics (e.g., RMS deviations or fidelity differences), making it difficult to assess the magnitude of the reported closeness.
No mention of error bars, statistical variation across runs, or sensitivity to numerical tolerances in the driven benchmarks or runtime measurements.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for highlighting the scope limitation in our benchmark. We agree that the quantitative comparisons rest on a single realistic parameter set and device, and we will revise the abstract and discussion to reflect this explicitly while preserving the illustrative value of the multi-metric evaluation.

read point-by-point responses

Referee: [Abstract] Abstract: the general recommendation for layered abstraction in pulse-level simulation rests on comparisons performed with a single fixed 'realistic parameter set' for one flux-tunable two-qubit device; no sweeps over anharmonicity, coupling strength, or flux range beyond the tested interval, nor cross-device validation, are reported, so the quantitative superiority of the Duffing model and the absence of effective-model effects in driven cases may be specific to the chosen values rather than generic.

Authors: We acknowledge the validity of this observation. The manuscript uses one representative parameter set chosen to match typical experimental values for a flux-tunable transmon pair with fixed coupler, enabling a consistent, multi-aspect benchmark (spectra, two-qubit terms, driven dynamics, leakage, runtime) on the same hardware model. This design isolates the effect of Hamiltonian abstraction level but does not include parameter sweeps or cross-device checks. We will revise the abstract to state that the results illustrate the relative fidelity of the models for this realistic case and support a layered strategy within the tested regime, rather than claiming generic quantitative superiority. The discussion section will be updated to note the limitation and the desirability of future sweeps. No new simulations are added, as they fall outside the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity; model comparisons use independent reference and fixed parameters

full rationale

The paper evaluates three distinct Hamiltonian models (effective two-level, Duffing, circuit-based transmon) against each other on a benchmark suite using one realistic parameter set for a flux-tunable device. The circuit-based model is treated as an external reference for comparison, with no derivations, fits, or predictions that reduce to the inputs by construction. No self-citations, ansatzes, or uniqueness claims are invoked in a load-bearing manner in the abstract or described content. The analysis is self-contained numerical comparison rather than tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on a chosen realistic parameter set for the device and the domain assumption that the three Hamiltonian descriptions are valid at their respective abstraction levels for the tested benchmarks.

free parameters (1)

realistic parameter set
Chosen numerical values representing the flux-tunable two-qubit device with fixed bus coupler; used as input for all three models.

axioms (1)

domain assumption The effective two-level, three-mode Duffing, and circuit-based transmon models are appropriate descriptions of the hardware at their respective levels of abstraction.
Invoked when using their outputs to determine relative fidelity and sufficiency for simulation objectives.

pith-pipeline@v0.9.0 · 5763 in / 1302 out tokens · 23723 ms · 2026-05-25T05:15:47.231247+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We compare three Hamiltonian descriptions... effective two-level model, a three-mode Duffing model, and a circuit-based transmon model... evaluate these models on a common benchmark suite spanning flux-dependent spectra, extracted two-qubit interaction terms, driven single-qubit dynamics, CZ gate dynamics, leakage...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Zijun Chen and et al. 2016. Measuring and Suppressing Quantum State Leakage in a Superconducting Qubit.Physical Review Letters116, 2 (2016), 020501. doi:10. 1103/PhysRevLett.116.020501

work page 2016
[2]

DiCarlo and et al

L. DiCarlo and et al. 2009. Demonstration of two-qubit algorithms with a superconducting quantum processor.Nature460, 7252 (2009), 240–244. doi:10.1038/nature08121

work page doi:10.1038/nature08121 2009
[3]

Preparation and Measurement of Three-Qubit Entanglement in a Superconducting Circuit

L. DiCarlo and et al. 2010. Preparation and Measurement of Three-Qubit En- tanglement in a Superconducting Circuit.Nature467, 7315 (2010), 574–578. arXiv:1004.4324 [cond-mat] doi:10.1038/nature09416

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nature09416 2010
[4]

Peter Groszkowski and Jens Koch. 2021. scqubits: a Python package for super- conducting qubits.Quantum5 (2021), 583. doi:10.22331/q-2021-11-17-583

work page doi:10.22331/q-2021-11-17-583 2021
[5]

Ling Jiang and et al. 2025. Microwave-activated two-qubit gates for fixed- coupling and fixed-frequency transmon qubits.Physical Review A111, 3 (2025), 032609

work page 2025
[6]

Jens Koch and et al. 2007. Charge-Insensitive Qubit Design Derived from the Cooper Pair Box.Physical Review A76, 4 (2007), 042319. doi:10.1103/PhysRevA. 76.042319

work page doi:10.1103/physreva 2007
[7]

Philip Krantz and et al. 2019. A Quantum Engineer’s Guide to Superconducting Qubits.Applied Physics Reviews6, 2 (2019), 021318. doi:10.1063/1.5089550

work page doi:10.1063/1.5089550 2019
[8]

Per-Olov Löwdin. 1970. On the Nonorthogonality Problem**The work reported in this paper has been sponsored in part by the Swedish Natural Science Research Council, in part by the Air Force Office of Scientific Research (OSR) through the European Office of Aerospace Research (OAR), U.S. Air Force under Grant AF-EOAR 67-50 with Uppsala University, and in pa...

work page doi:10.1016/s0065-3276(08)60339-1 1970
[9]

Majer and et al

J. Majer and et al. 2007. Coupling superconducting qubits via a cavity bus.Nature 449, 7161 (2007), 443–447. doi:10.1038/nature06184

work page doi:10.1038/nature06184 2007
[10]

2024.Building quantum computers: a practical introduction

Shayan Majidy, Christopher Wilson, and Raymond Laflamme. 2024.Building quantum computers: a practical introduction. Cambridge University Press

work page 2024
[11]

Physical Review Letters 85(10), 2200–2203 (2000)

V. Negîrneac and et al. 2021. High-Fidelity Controlled-Z Gate with Maximal In- termediate Leakage Operating at the Speed Limit in a Superconducting Quantum Processor.Physical Review Letters126, 22 (2021), 220502. doi:10.1103/PhysRevLett. 126.220502

work page doi:10.1103/physrevlett 2021
[12]

Daniel Puzzuoli and et al. 2023. Qiskit Dynamics: A Python package for simulat- ing the time dynamics of quantum systems.Journal of Open Source Software8, 90 (2023), 5853. doi:10.21105/joss.05853

work page doi:10.21105/joss.05853 2023
[13]

Qiskit Development Team. 2025. Qiskit Dynamics documentation. https://qiskit- community.github.io/qiskit-dynamics/. Accessed 2026-04-18

work page 2025
[14]

Qiskit Development Team. 2025. Simulating backends at the pulse-level with Dy- namicsBackend. https://qiskit-community.github.io/qiskit-dynamics/tutorials/ dynamics_backend.html. Qiskit Dynamics documentation; accessed 2026-04-18

work page 2025

[1] [1]

Zijun Chen and et al. 2016. Measuring and Suppressing Quantum State Leakage in a Superconducting Qubit.Physical Review Letters116, 2 (2016), 020501. doi:10. 1103/PhysRevLett.116.020501

work page 2016

[2] [2]

DiCarlo and et al

L. DiCarlo and et al. 2009. Demonstration of two-qubit algorithms with a superconducting quantum processor.Nature460, 7252 (2009), 240–244. doi:10.1038/nature08121

work page doi:10.1038/nature08121 2009

[3] [3]

Preparation and Measurement of Three-Qubit Entanglement in a Superconducting Circuit

L. DiCarlo and et al. 2010. Preparation and Measurement of Three-Qubit En- tanglement in a Superconducting Circuit.Nature467, 7315 (2010), 574–578. arXiv:1004.4324 [cond-mat] doi:10.1038/nature09416

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/nature09416 2010

[4] [4]

Peter Groszkowski and Jens Koch. 2021. scqubits: a Python package for super- conducting qubits.Quantum5 (2021), 583. doi:10.22331/q-2021-11-17-583

work page doi:10.22331/q-2021-11-17-583 2021

[5] [5]

Ling Jiang and et al. 2025. Microwave-activated two-qubit gates for fixed- coupling and fixed-frequency transmon qubits.Physical Review A111, 3 (2025), 032609

work page 2025

[6] [6]

Jens Koch and et al. 2007. Charge-Insensitive Qubit Design Derived from the Cooper Pair Box.Physical Review A76, 4 (2007), 042319. doi:10.1103/PhysRevA. 76.042319

work page doi:10.1103/physreva 2007

[7] [7]

Philip Krantz and et al. 2019. A Quantum Engineer’s Guide to Superconducting Qubits.Applied Physics Reviews6, 2 (2019), 021318. doi:10.1063/1.5089550

work page doi:10.1063/1.5089550 2019

[8] [8]

Per-Olov Löwdin. 1970. On the Nonorthogonality Problem**The work reported in this paper has been sponsored in part by the Swedish Natural Science Research Council, in part by the Air Force Office of Scientific Research (OSR) through the European Office of Aerospace Research (OAR), U.S. Air Force under Grant AF-EOAR 67-50 with Uppsala University, and in pa...

work page doi:10.1016/s0065-3276(08)60339-1 1970

[9] [9]

Majer and et al

J. Majer and et al. 2007. Coupling superconducting qubits via a cavity bus.Nature 449, 7161 (2007), 443–447. doi:10.1038/nature06184

work page doi:10.1038/nature06184 2007

[10] [10]

2024.Building quantum computers: a practical introduction

Shayan Majidy, Christopher Wilson, and Raymond Laflamme. 2024.Building quantum computers: a practical introduction. Cambridge University Press

work page 2024

[11] [11]

Physical Review Letters 85(10), 2200–2203 (2000)

V. Negîrneac and et al. 2021. High-Fidelity Controlled-Z Gate with Maximal In- termediate Leakage Operating at the Speed Limit in a Superconducting Quantum Processor.Physical Review Letters126, 22 (2021), 220502. doi:10.1103/PhysRevLett. 126.220502

work page doi:10.1103/physrevlett 2021

[12] [12]

Daniel Puzzuoli and et al. 2023. Qiskit Dynamics: A Python package for simulat- ing the time dynamics of quantum systems.Journal of Open Source Software8, 90 (2023), 5853. doi:10.21105/joss.05853

work page doi:10.21105/joss.05853 2023

[13] [13]

Qiskit Development Team. 2025. Qiskit Dynamics documentation. https://qiskit- community.github.io/qiskit-dynamics/. Accessed 2026-04-18

work page 2025

[14] [14]

Qiskit Development Team. 2025. Simulating backends at the pulse-level with Dy- namicsBackend. https://qiskit-community.github.io/qiskit-dynamics/tutorials/ dynamics_backend.html. Qiskit Dynamics documentation; accessed 2026-04-18

work page 2025