Nonlinear Feedback in Josephson Circuit Optimization: Application to a Kerr-Reversal JTWPA

arXiv:2610.10715 · quant-ph · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Nonlinear Feedback in Josephson Circuit Optimization".

Kai: The gist The Josephson Circuits Optimizer addresses computational demands in optimizing Josephson-based nonlinear microwave devices by using harmonic balance simulations in two stages,

Mira: First, who's behind it and why it matters.

Paper summary: Kai: So, we're looking at this paper called "Nonlinear Feedback in Josephson Circuit Optimization: Application to a Kerr-Reversal JTWPA". Basically, they're tackling the computational headache of finding the best settings for these Josephson-based nonlinear microwave devices.

Mira: It seems like their main idea is to use harmonic balance simulations in two stages to handle this problem. They claim this approach improves optimization accuracy under pump-induced effects.

Kai: Right, so they start with fast linear simulations to pick good circuit configurations based on things like impedance and phase matching first, and then they use nonlinear simulations on those candidates for the operating conditions.

Lev: That sounds like it could be a way to manage the computational cost since full nonlinear sims are expensive across a whole parameter space.

Kai: Exactly, and there's this nonlinear feedback extension they introduce to make that initial linear stage better when you have pump effects messing things up.

Mira: The paper suggests that pump-induced stuff, like changes in impedance or modifications to phase matching because of Kerr effects, isn't captured in the first linear stage exploration.

Lev: That makes sense from a hardware standpoint; if the underlying physics is changing during the optimization loop, you need a way to account for that change.

Kai: They use this information from the nonlinear response to update that metric before starting the next iteration of optimization. This update is represented by this correction term, C, which modifies the metric M as M(n+one) = g M(n) <ref:2610.10715#pg1>.

Mira: The paper shows they evaluate their framework using a lumped-element model of a 4WM JTWPA from Silent Waves <ref:2610.10715#pg1>. This model mimics the reversed-Kerr architecture shown in figure two <ref:2610.10715#pg1>.

Kai: And they benchmark this against experimental characterization, where the nonlinear harmonic balance simulations reproduce key experimental features, like how the gain profile changes with pump conditions.

Lev: If you're running this on real hardware, you'd be checking if these simulated results actually match what you measure when you pump it.

Kai: They focus on constructing this feedback correction term by varying parameters like the critical current I0 and ground capacitance Cg to define a candidate observable for that correction.

Mira: For selecting the initial circuit configuration, they use a metric based on the input reflection coefficient S11, defined as M = <S11,dB(f) five–nine GHz, S11,dB(3fp) > -zero point zero five dB.

Kai: That first part of their linear exploration measures the average input reflection within the signal band to show improved impedance matching across a range of frequencies.

Lev: So they're using a reflection coefficient as the metric to guide where they look in that massive parameter space P.

Kai: The initial circuit configuration, p*, is selected entirely from this linear exploration by minimizing that linear metric Mlin(I0, Cg) = M Slin eleven (I0, Cg) <ref:2610.10715#pg3>.

Paper summary: Mira: Then they take that selected configuration and run the nonlinear harmonic balance simulations over the working point space Q to find the optimal operating point q*.

Kai: The performance function F used here is something like the average gain, which helps determine what the best operating conditions are for that specific circuit.

Lev: So you pick a good structure linearly, then you fine-tune how it's running nonlinearly. That's a standard optimization split.

Kai: But they then look at the metric M*, which is evaluated from the pumped response for that selected configuration, and they quantify its departure from the linear prediction using delta M*.

Mira: This pointwise linear-to-nonlinear discrepancy, delta M*, shows how much that metric changes under pumped operation and it's not uniform across different circuit configurations.

Lev: If delta M* depends on the configuration, that tells you something important about where the real performance peaks are hiding relative to what a simple linear model predicts.

Kai: The paper concludes that varying the reflection metric contains information about amplifier performance that a linear response alone doesn't capture.

Mira: They point out that the maximum-gain configuration differs from the linear optimum and lives in a region where this discrepancy delta M* is pronounced.

Lev: So what does this mean practically for someone who just listens to the show? It means you can't trust just picking the best structure based on simple matching; you have to look at how much things shift when they start pumping power.

Kai: The next step, which they suggest, is investigating delta M* in subsequent optimization cycles where this quantity comes from nonlinear simulation of only the selected configuration.

Mira: They plan to quantitatively assess the relationship between this metric discrepancy and actual amplifier performance and define a specific update rule for the feedback term C.

Lev: That moves it from just finding an optimum to creating a smarter, self-correcting optimization loop that accounts for those nonlinear pump effects directly into the search process.

Kai: So, to wrap up this discussion on "Nonlinear Feedback in Josephson Circuit Optimization: Application to a Kerr-Reversal JTWPA", they are proposing a way to blend fast linear configuration selection with nonlinear feedback corrections during the optimization process.

Mira: The authors claim this method helps optimize Josephson-based nonlinear microwave devices more effectively by using harmonic balance simulations in two stages.

Lev: It seems like a practical approach for getting better results on these types of complex superconducting circuits when you're trying to reach high performance levels.

Kai: Their work is important because it shows that the simple linear metric isn't enough when pump effects are involved, and it points toward how to incorporate that nonlinear shift into the optimization itself.

Mira: The implications are in how we design these microwave devices for applications where they need to operate under real-world pumping conditions, like in quantum computing setups.

Conclusion: Kai: So we've been looking at how this paper tackles finding the best settings for these Josephson devices using harmonic balance simulations in two stages, right?

Mira: Yeah, it’s about taking that optimization problem and adding a nonlinear feedback extension to make the whole process more accurate when pump effects are pushing things around.

Kai: It's titled "Nonlinear Feedback in Josephson Circuit Optimization: Application to a Kerr-Reversal JTWPA," and the authors are working with these superconducting circuits.

Mira: The core idea is using this two-stage method—linear simulation first, then nonlinear analysis—to figure out the best circuit configuration and its operating point simultaneously.

Lev: From my side, it’s interesting because real hardware doesn't always follow those smooth linear predictions when you start pumping power in.

Kai: Exactly, and they found that this nonlinear feedback helps update the metric used for the next linear step based on what the nonlinear simulation actually shows.

Mira: They show that this approach lets you find a maximum-gain configuration that might be completely different from what a purely linear search would pick.

Kai: It means you can't just rely on one simple linear scan to find the best amplifier settings; you need to account for those pump-induced changes in the device's behavior.

Lev: So, they’re suggesting that for real quantum hardware, you can use this nonlinear discrepancy information—that delta M—to guide the search more intelligently.

Mira: That points toward a way to make the optimization loop itself aware of those nonlinear physics during the design process.

Kai: It suggests that finding a high-performance device isn't just about hitting a single linear sweet spot; it’s about navigating this shift in performance under operating conditions.

Lev: And what they are doing with that delta M, as they suggest, is getting more rigorous on how much that specific nonlinear change actually impacts the final amplifier gain and stability.

Emanuele Palumbo, Alessandro Alocco, Andrea Celotto, Luca Fasolo, Bernardo Galvano, Patrizia Livreri, Emanuele Enrico

Department of Applied Science and Technology, Politecnico di Torino Department of Engineering, University of Palermo Institute Nazionale di Ricerca Metrologica

quant-ph

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 4 pages, 4 figures

Code: https://github.com/SQE-INRiM/JosephsonCircuitsOptimizer

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 76/100

The gist: The gist The Josephson Circuits Optimizer addresses computational demands in optimizing Josephson-based nonlinear microwave devices by using harmonic balance simulations in two stages, introducing a

Key concepts

Optimization Framework: This is a two-stage process to find the best circuit design and operating conditions. First, it uses fast linear simulations to find a good circuit layout. Then, it uses slower nonlinear harmonic balance simulations on that layout to find the best operating point under pump conditions.
Linear Metric (M): This is a measurable quantity based on input reflection ($S_{11}$) within a specific frequency band. It's used in the initial stage to quickly select promising circuit designs by minimizing this metric, aiming for better impedance matching.
Pointwise Linear-to-Nonlinear Discrepancy ($ΔM⨁): This measures how much the performance metric changes when moving from a linear prediction to a nonlinear simulation. It quantifies how strong pump effects modify the metric, showing that the optimal configuration under pumping differs significantly from the linear optimum.
Harmonic Balance Simulation: This is a powerful numerical technique used to model and analyze nonlinear circuits. It allows researchers to accurately simulate complex device behavior, especially under strong pump conditions, by solving equations related to the circuit's response.

Terminology

Trace — 4× in this paper · explained in 2 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Josephson Circuits Optimizer — 1× in this paper

Summary

The gist The Josephson Circuits Optimizer addresses computational demands in optimizing Josephson-based nonlinear microwave devices by using harmonic balance simulations in two stages, introducing a nonlinear feedback extension to improve optimization accuracy under pump-induced effects.

Optimization Framework

The original JCO framework separates the optimization of the circuit configuration from the optimization of its operating conditions First, the device parameter space P is explored through computationally efficient linear simulations using a metric M, yielding the optimal circuit configuration p∗ = arg min p∈P M(S(p)) The selected configuration p∗ is then analyzed through nonlinear harmonic balance simulations over the workingpoint space Q to determine the optimal operating point q∗ = arg max q∈Q F(X(q; p∗)) Because the linear optimization metric may no longer remain representative under strong pumping, information extracted from the nonlinear response is used to update it before the next iteration The nonlinear correction is represented by the correction term C, which updates the metric as M(n+1) = g M(n), C(n) This iterative procedure continues until successive corrections become negligible.

Device Model and Validation

The proposed framework is evaluated using the lumped-element model of a 4WM JTWPA from Silent Waves [12] This equivalent circuit reproduces the reversed-Kerr architecture, which is shown in Fig. 2. The model is benchmarked against experimental characterization, where nonlinear harmonic balance simulations reproduce the main experimental features, including the evolution of the gain profile with pump conditions.

Linear Metric and Configuration Selection

The analysis investigates a candidate observable for constructing the feedback correction term by varying parameters like critical current I0 and ground capacitance Cg. Each circuit configuration is evaluated through a metric based on the input reflection coefficient S11, defined as M = ⟨S11,dB(f)⟩ 5–9 GHz, S11,dB(3fp) > −0.05 dB. The first term measures the average input reflection within the signal band to indicate improved impedance matching. The initial circuit configuration is selected entirely from this linear exploration by minimizing the metric Mlin(I0, Cg) = M Slin 11 (I0, Cg).

Pointwise Linear-to-Nonlinear Discrepancy

In the proposed feedback workflow, nonlinear simulation is performed only for the configuration selected by the linear stage. The corresponding metric at this selected configuration, evaluated from the pumped response, is M∗nonlin = M Snonlin 11 I∗0, C∗g, and its departure from the linear prediction is quantified by ∆M∗ = M∗nonlin − Mlin I∗0, C∗g. This pointwise linear-to-nonlinear discrepancy quantifies the modification of the metric under pumped operation and serves as a candidate input for constructing the feedback correction term C. The analysis shows that ∆M exhibits a pronounced dependence on the circuit configuration, indicating that the pumped response does not result in a uniform modification of the linear metric.

Conclusion and Future Work

The work demonstrates that the variation of the reflection metric contains information related to amplifier performance not captured by linear response alone. The maximum-gain configuration differs from the linear optimum and lies in a region characterized by a pronounced metric discrepancy. This motivates the investigation of ∆M∗ in subsequent optimization cycles, where this quantity can be obtained from nonlinear simulation of only the selected configuration. Future work will quantitatively assess the relation between the metric discrepancy and amplifier performance and define a corresponding update rule.

ACKNOWLEDGMENT

The authors gratefully acknowledge Silent Waves for providing the Josephson Traveling-Wave Parametric Amplifier used in this work.

REFERENCES

[1] L. Fasolo, A. Greco, and E. Enrico, “Superconducting Josephson-based metamaterials for quantum-limited parametric amplification: A review,” in Advances in Condensed-Matter and Materials Physics, IntechOpen, 2019

[2] M. Esposito, A. Ranadive, L. Planat, and N. Roch, “Perspective on traveling wave microwave parametric amplifiers,” Appl. Phys. Lett., vol. 119, Art. no 120501, 2021

[3] C. Macklin et al., “A near-quantum-limited Josephson traveling-wave parametric amplifier,” Science, vol. 350, pp. 307–310, 2015

[4] K. Peng, R. Poore, P. Krantz, D. E. Root, and K. P. O’Brien, “Xparameter based design and simulation of Josephson traveling-wave parametric amplifiers for quantum computing applications,” in Proc IEEE Int Conf Quantum Comput Eng (QCE), 2022

[5] D. Shiri et al., “Modeling and harmonic balance analysis of superconducting parametric amplifiers for qubit readout: A tutorial,” IEEE Microw Mag, vol. 25, pp. 54–73, 2024

[6] E. Palumbo et al., “JCO: Optimization framework for nonlinear superconducting circuits using a lumped-element approach and harmonic balance,” IEEE Trans Appl Supercond, vol. 36, pp. 1–5, 2026

[7] E. Palumbo, “JosephsonCircuitsOptimizer,” GitHub repository

[8] K. P. O’Brien, “JosephsonCircuits.jl documentation,” Online Available: https://josephsoncircuits.org/stable/

[9] K. P. O’Brien, C. Macklin, I Siddiqi, and X Zhang, “Resonant phase matching of Josephson junction traveling wave parametric amplifiers,” Phys Rev Lett., vol 113, Art no 157001, 2014

[10] S. Kern et al., “Reflection-enhanced gain in traveling-wave parametric amplifiers,” Phys Rev B, vol. 107, Art no 174520, 2023

[11] K. Peng, M Naghiloo, J Wang, G D Cunningham, Y Ye, and K P O’Brien, “Floquet-mode traveling-wave parametric amplifiers,” PRX Quantum, vol 3, Art no 020306, 2022 doi: 10.1103/PRXQuantum.3.020306

[12] A Ranadive et al., “Kerr reversal in Josephson meta-material and traveling wave parametric amplification,” Nat Commun, vol 13, Art no 1737, 2022

[13] M T Bell, “Josephson traveling-wave parametric amplifier with inverse Kerr phase matching,” Phys Rev Applied, vol. 25, Art no 014075, 2026 doi: 10.1103/frfk-ty7x

[14] A Yu Levochkina et al., “Investigating pump harmonics generation in a SNAIL-based traveling wave parametric amplifier,” Supercond Sci Technol, vol 37, Art no 115021, 2024

[15] E Palumbo, “Data for ‘Nonlinear Feedback in Josephson Circuit Optimization: Application to a Kerr-Reversal JTWPA’, Zenodo, 2026 doi: 10.

Improvements for AI systems

textbfOutput of Optimized Circuit Configurations During Pumping Cycles: The improved AI system can dynamically update its circuit design based on pumped response information by using the formula M(n+1) = g M(n), C(n) to refine the metric for subsequent optimization cycles, effectively allowing the system to converge towards configurations that are optimal under operating conditions rather than just linear predictions.
textbfDiscovery of Non-Uniform Metric Modification: The system can identify regions where the pumped response does not result in a uniform modification of the linear metric, specifically by analyzing the discrepancy function ∆M(I0, Cg) = Mnonlin(I0, Cg) − Mlin(I0, Cg), which highlights configuration-dependent performance shifts.
textbfPerformance Mapping Under Nonlinear Operation: The improved system can map out how the maximum-gain configuration differs from the linear optimum and lies in a region characterized by a pronounced metric discrepancy, enabling researchers to pinpoint exactly where nonlinear effects lead to superior performance, as seen when comparing the red marker (linear optimum) with the green marker (maximum average gain configuration).
textbfEfficient Feedback Implementation: The AI can implement a feedback loop that can be obtained from the nonlinear simulation of only the configuration selected by the linear stage, preserving the computational advantage of predominantly linear parameter-space exploration, ensuring that expensive nonlinear simulations are used judiciously to guide iterative optimization.

Abstract

Optimizing Josephson-based nonlinear microwave devices is computationally demanding because the straightforward approach requires exploring broad circuit parameter spaces through expensive nonlinear simulations. Josephson Circuits Optimizer addresses this problem by using harmonic balance simulations in two stages: fast linear simulations to select promising circuit configurations according to properties such as impedance and phase matching, followed by nonlinear simulations to optimize their operating conditions. However, pump-induced effects such as impedance renormalization and Kerr-induced modifications of phase matching are not captured during the initial linear stage. We therefore introduce a nonlinear feedback extension that transfers information from the pumped response back to the linear optimization. The approach is investigated using a Josephson traveling-wave parametric amplifier with a reversed-Kerr architecture, whose model is validated against experimental gain measurements. We then evaluate the difference between the linear and pumped response values of a metric based on the input reflection coefficient, while excluding configurations with insufficient third-harmonic suppression. This discrepancy is considered a candidate feedback observable, as it exhibits features similar to the gain landscape, supporting its use in subsequent optimization cycles.