Modelling Enclosures for Large-Scale Superconducting Quantum Circuits

B. Vlastakis; P. A. Spring; P. J. Leek; T. Tsunoda

REVIEW 3 major objections 6 minor 36 references

Modelling Enclosures for Large-Scale Superconducting Quantum Circuits

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper predicts that periodic arrays of inductive shunts make cavity-mediated inter-qubit coupling and drive-line crosstalk decay exponentially with distance, and it validates the prediction with finite-element simulation of a…

desk verdict Useful and mostly solid engineering-physics paper: the new mode-frequency models are worth refereeing, but the headline crosstalk decay rate is not yet independently validated. read the letter →

arxiv 1909.02104 v2 pith:5IQWCSEG submitted 2019-09-04 quant-ph cond-mat.mes-hallphysics.app-ph

P. A. Spring , T. Tsunoda , B. Vlastakis , P. J. Leek This is my paper · ORCID

classification quant-phcond-mat.mes-hallphysics.app-ph

keywords superconductingqubitsenclosuremodesinductiveshuntarraysthrough-substrateviascrosstalksuppressionplasmamodelevanescentquantumprocessorscaling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper predicts that a superconducting circuit in a conducting enclosure loaded with a periodic array of inductive shunts—thin pillars or vias connecting the two faces—will have all enclosure-mediated crosstalk fall off exponentially with distance, for enclosures of arbitrary size. The authors derive two closed-form models for the enclosure modes: an anisotropic-plasma model for thin shunts and a coupled-cavity circuit model for thick shunts. Below the cutoff frequency set by the shunt array, qubit-coupling and drive-line crosstalk are carried by an evanescent cylindrical mode with the spatial law $J_{ij}\propto K_0(d_{ij}/\delta_p)$, so distant qubits are effectively isolated. A finite-element simulation of a device with a $21\times21$ grid of qubits matches this decay. If the prediction holds, this removes one scaling obstacle for large monolithic superconducting quantum processors.

What carries the argument

The load-bearing object is the plasma model of the periodic shunt array. The array of thin conducting cylinders is treated as an anisotropic medium with relative permittivity $\epsilon_p(f)=1-(f_p/f)^2$ along the cylinder axis, with plasma frequency $f_p=f_a/\sqrt{\pi(\ln(a/r)-\Pi)}$ set by the radius $r$ and spacing $a$. Mode frequencies of the shunted enclosure are $f'_{nm}=\sqrt{f_{nm}^2+f_p^2}$, and below $f_p$ the dominant $TM_{00}$ radial mode becomes evanescent, with penetration depth $\delta_p=1/\sqrt{\epsilon_0\epsilon_r\mu_0(\omega_p+\omega_q)(\omega_p-\omega_q)}$ and a $K_0(d/\delta_p)$ spatial profile. For thick shunts, a circuit model maps the two-dimensional array of magnetically coupled cavities onto a one-dimensional chain through a Kronecker sum, producing a quadratic band edge and a cutoff $f_c=f_0/\sqrt{1+8\beta}$; the two models converge on the same exponential-crosstalk result.

What would settle it

Compute the plasma penetration depth from the geometry-only formulas (eqs. (4) and (18)) for a thick-shunt case such as $r/a=0.25$, rerun the finite-element crosstalk simulation without feeding in the simulated fundamental frequency, and compare the fitted decay length to that geometry-only value; a significant mismatch would falsify the claim that the model predicts the exponential length scale in the circuit-model regime.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that an enclosure loaded with a periodic inductive-shunt array has a cutoff frequency below which no propagating modes exist, so qubits below cutoff interact only through evanescent fields. The transverse exchange coupling between qubits $i$ and $j$ is $J_{ij}=2g^2\,\frac{\omega_q}{(v/\delta_0)^2}K_0(d_{ij}/\delta_p)$, and the drive-line-to-qubit coupling is $\varepsilon_{ij}=\varepsilon_0 K_0(d_{ij}/\delta_p)$, where $K_0$ is the modified Bessel function of the second kind. For separations $d_{ij}\gg\delta_p$, $K_0(x)\sim\sqrt{\pi/2}\,e^{-x}/\sqrt{x}$, so both interactions are exponentially small beyond a few penetration depths $\delta_p$. The authors further show that the same spatial law follows from a bound-state picture of a qubit below a two-dimensional quadratic band edge, and they verify the decay against finite-element simulation of an enclosure containing 441 qubits.

Load-bearing premise

The quantitative support for the thick-shunt case uses the simulated fundamental cavity frequency as an input to the model rather than computing it from the model's own geometry-only formula; if that substitution masks a breakdown in the formula, the measured exponential decay length has not actually been predicted.

Editorial extensions

If this is right

Enclosure-mediated qubit couplings become local: qubits separated by several plasma penetration depths have negligible $J_{ij}$, so large processors are not plagued by long-range cavity-mediated interactions.
Drive-line crosstalk to distant qubits is exponentially suppressed, allowing microwave control lines to be routed without coupling to far-away qubits.
Shunt radius and spacing set both the cutoff frequency and the decay length through eq. (18), giving designers a direct tuning knob for locality.
Because the fundamental enclosure frequency is pushed above qubit frequencies, qubits are not immersed in a dense continuum of enclosure modes, reducing radiative relaxation and spurious coupling.
The finite-element result for a 441-qubit enclosure shows the approach working at the scale of current noisy intermediate-scale devices, and the scale-independent cutoff supports extending it to larger arrays.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the same $K_0$ evanescent law should apply to any element coupled to the enclosure's $TM_{00}$ mode, including readout resonators and tunable couplers, so crosstalk budgets could be estimated without full-wave simulation.
Beyond the paper: the bound-state equivalence suggests a spectroscopic route to measure $\delta_p$ directly—detune a probe qubit below cutoff and extract the band-edge curvature from its frequency shift, bypassing antenna-coupling uncertainties.
Beyond the paper: if the exponential locality holds, layout designers could partition a processor into local clusters separated by a few $\delta_p$, keeping long-range interactions on dedicated buses rather than through the enclosure.
Beyond the paper: a direct experimental check would be a two-port transmission measurement between antennas in a shunted enclosure below cutoff; the attenuation length should be $\delta_p$ from geometry, independent of the antenna details.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful paper with one genuinely new modeling result and one real but fixable validation gap. The cleanest new thing is the 2D circuit model and its exact reduction to a 1D impedance matrix (eq. 29). That mapping is neat and gives closed-form mode frequencies (eq. 8) for the Lb = 0, Lg, 2Lg cases, while also reproducing the tight-binding limit. The plasma formula eq. (6) is not new—Murray and Abraham used it empirically—but the paper credits that, and it verifies eq. (6) against HFSS for r/a < 0.1. For r/a > 0.1, the circuit model with beta and beta1 matches the FE spectrum well, including the cutoff and the band gaps. That part is solid and directly useful.

The soft spot is the headline claim of exponentially suppressed crosstalk. The K0 form in eqs. (14)–(16) is a well-motivated prediction, and the FE data do show exponential decay. But the quantitative length scale is not independently predicted. In Section IV the paper uses the FE-simulated fundamental mode for the plasma frequency rather than eq. (4), and in Fig. 6(c,d) delta_p is also fitted to the same FE crosstalk data. So what is validated is the shape, not the parameter-free decay rate. That matters because eq. (4) is the only geometric predictor for r/a < 0.1, and the claim about arbitrarily large circuits leans on it. The single 21x21 geometry does not by itself establish size independence, and there is no experimental data. None of this is fatal; the fix is straightforward. I would want a comparison using eq. (18) without feeding in the simulated fundamental mode, and at least one additional device size. The bound-state route in Section III also gives an independent consistency check via eqs. (19)–(21), and that is underused.

Who gets value from this: device engineers designing shunted packages, and anyone working on mode engineering for large superconducting processors. I would send it to referees. The mode-frequency models alone justify serious review, and the crosstalk section is clearly written enough that the missing test is easy to specify. I would cite the mode-frequency results now; I would hold off on citing the 'arbitrarily large' claim until the decay rate is shown to come out of the geometry without fitting to the same simulation. Worth a reading-group slot too—there is a good discussion to be had about circular validation.

Referee Report

3 major / 6 minor

Summary. The manuscript develops two analytical descriptions of the electromagnetic environment of a superconducting circuit enclosed in a cavity that is inductively shunted by a periodic square array of conducting cylinders. The first is a plasma model, valid for shunt radius-to-spacing ratio r/a < 0.1, which yields mode frequencies f'_nm = sqrt(f_nm^2 + f_p^2) and a plasma-penetration-depth prediction for cross-talk decay. The second is a circuit (tight-binding-like) model for r/a > 0.1, giving the mode spectrum in terms of an inductive coupling parameter beta. Using these models, the authors derive closed-form expressions for cavity-mediated inter-qubit couplings J_ij and drive-line cross-talk epsilon_ij, both proportional to the modified Bessel function K0(d_ij/delta_p), hence exponentially suppressed at distances large compared with delta_p. The predictions are compared with HFSS finite-element simulations, including eigenmode frequencies and cross-talk in a 21x21 qubit device, and the paper concludes that such inductively shunted enclosures can support arbitrarily large superconducting circuits with local cross-talk.

Significance. If the quantitative predictions are independently validated, the paper makes a genuinely useful design statement: enclosure-mediated cross-talk in shunted packages is not a long-range problem, and the relevant length scale is a closed-form function of shunt geometry and qubit frequency. The analytical derivations in Sections II and III, particularly the plasma-model dispersion, the circuit-model mapping in Appendix B, and the transverse-coupling derivation in Appendix C, are substantial and clearly presented. The mode-frequency comparisons against FE simulations are a real strength of the paper, especially the systematic study of r/a in Fig. 2 and the circuit-model comparison in Fig. 4. However, as detailed in the major comments, the quantitative validation of the cross-talk decay length, which is the most novel and load-bearing claim, is incomplete: the decay rate is either taken from the FE eigenmode simulation or fitted to the same cross-talk data, so the paper does not yet demonstrate that eq. (18) predicts the observed length scale.

major comments (3)

[Section IV, paragraph after Eq. (24)] The quantitative cross-talk validation uses the fundamental cavity frequency from the FE eigenmode simulation (Table I) as the plasma frequency in Eq. (15), rather than the analytically predicted fp from Eq. (4). This is stated explicitly in the text. Consequently, Figs. 6(a) and 6(b) validate the exponential/K0 form of the decay, but they do not test the parameter-free prediction of the decay length delta_p(r,a) given by Eq. (18). Since Eq. (4) is the only closed-form link between shunt geometry and delta_p, the central claim that the decay rate is a simple function of shunt radius and spacing is not tested even in the r/a < 0.1 regime, where the substitution could have been made directly. Please repeat the comparison using fp from Eq. (4) (with the effective permittivity of Appendix A), and show the resulting agreement or disagreement.
[Fig. 6(c,d) and surrounding text] In Figs. 6(c) and 6(d), the blue dots are obtained by fitting delta_p in Eq. (14) to the same FE Gamma_Q cross-talk data that the model is meant to explain. Using the same data both to fit the sole free parameter and to claim validation is circular: it demonstrates only that K0(d/delta_p) is a good fitting function, not that delta_p is predicted by Eqs. (15) and (18). The fitted values should be compared with the analytic prediction computed from Eq. (4), or from an independently determined fp, without fitting any cross-talk data; the comparison should be shown in the figure or discussed in the text.
[Section II.C and Eq. (21)] For r/a > 0.1, the paper argues that the shunt array is most effective, yet the circuit-model decay length Eq. (21) depends on the coupling parameter beta, which in Fig. 4 is fitted to FE mode frequencies; no geometric formula for beta is provided. Thus, in the r/a > 0.1 regime the paper does not deliver a parameter-free geometric prediction of the cross-talk length scale, but rather a functional form with one fitted parameter. The conclusion that the plasma model predicts the rate of cross-talk decay as a simple function of shunt radius and spacing (Section V) is therefore not supported in this regime. Please state this limitation explicitly, or derive beta (or equivalently delta_b) from geometry and compare it with the simulated decay.

minor comments (6)

[Section I, Introduction] There is a typo: 'simple contigious cavity enclosure' should be 'simple contiguous cavity enclosure'.
[Fig. 1 caption] The caption contains a duplicated word: 'decreasing the the cavity mode frequency' should read 'decreasing the cavity mode frequency'.
[Fig. 2 caption] The caption reads 'a cavity containing a containing an inductive shunt array'; the duplicated 'containing' should be removed.
[Eqs. (1), (8), and Fig. 4] The symbol n is used both for a mode index in Eq. (1) and for the number of cells in the circuit-model spectrum of Eq. (8) and Fig. 4. This is confusing; please use a different symbol for one of the two meanings.
[Section IV and Table I] The statement that Eq. (6) 'rapidly diverges' for r/a > 0.1 would be more informative with a quantitative error metric for each radius, for example the normalized relative error shown in Fig. 2, which is currently absent from Table I.
[Abstract and Section V] The claim that the approach works for 'arbitrarily large quantum circuits' is extrapolated from one 21x21 FE geometry; the analytical models are infinite-array arguments, and finite-size effects are not studied. I suggest explicitly framing the scalability statement as an extrapolation supported by the periodic-array models, or adding a second, differently sized simulation to demonstrate size independence.

Circularity Check

2 steps flagged · score 4.0 of 10

The analytic exponential-crosstalk derivation is self-contained, but the device-scale validation of the decay rate is partly circular: the plasma frequency/penetration depth is taken from the same FE eigenmode simulation or fitted to the same FE crosstalk data, not computed from eq. (18).

fitted input called prediction [Section IV, paragraph beginning 'When calculating the plasma penetration depth']
"When calculating the plasma penetration depth (eq. 15), we used the fundamental cavity frequency found from eigenmode simulation (Table I) as the plasma frequency, rather than that from eq. (4)."

The model's geometric prediction for the crosstalk decay length is eq. (18), which depends only on r and a through eq. (4). In the only device-scale test, f_p is instead taken from the same HFSS eigenmode simulation whose crosstalk the model is meant to predict. Consequently the agreement in Fig. 6(a,b) does not test the parameter-free prediction of the decay rate from geometry: the exponential length scale is imported from the FE model rather than derived from eq. (18). The functional (K0/exponential) form is tested, but the quantitative decay rate used for the 'arbitrarily large circuits' claim is not independently predicted.
fitted input called prediction [Fig. 6 caption, panels (c) and (d)]
"Red crosses are calculated using eq. (15) and blue dots are found by fitting eq. (14) to the FE simulation ΓQ values, with δp as the sole fit parameter."

Here the decay length δp is fitted directly to the FE simulation's crosstalk data, so the blue dots cannot independently validate the model: they are constructed from the data they are compared with. At most the plot shows consistency between the fitted length and the δp obtained from the FE eigenmode frequency. Neither curve is generated from the geometric formula eq. (18), so the paper's claim that eq. (18) predicts the rate of cross-talk decay from shunt radius and spacing is not actually demonstrated. The disclosure of the fit is transparent, but the 'prediction' of the quantitative decay rate is partly circular.

full rationale

The central analytical derivation, eqs. (14) and (16), follows from a standard plasma/waveguide evanescent-mode picture and the trans-impedance calculation in Appendix C; there is no equation-level identity that makes the crosstalk formula equal to its inputs. The exponential/K0 functional form is derived independently and the plasma-model mode-frequency calculations in Figs. 2 and 4 are substantive comparisons against FE simulations. The circularity concern is confined to the Section IV validation of the crosstalk decay rate: the paper explicitly replaces f_p from eq. (4) with the FE eigenmode fundamental frequency, and Fig. 6(c,d) additionally fits δp to the same FE crosstalk data. Therefore the quantitative decay length is not predicted from first principles in the validating test, even though the exponential shape is. This is a partial, disclosed validation circularity rather than a fully constructed equivalence: the predicted curves in Fig. 6(a,b) use a separately simulated eigenmode quantity, not a direct fit to the plotted ΓQ data. The self-citations to the group's architecture papers are not load-bearing for the core derivation. Overall the model has independent content, but the paper's strongest quantitative claim about the decay rate is under-supported by the validation as written.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The central quantitative prediction depends on effective-medium and circuit-model assumptions, and for the realistic device it uses fp taken from the FE simulation and delta_p fitted to it, so the independent content of eq. (18) is not demonstrated in the device validation.

free parameters (4)

beta = fitted to FE mode frequencies; e.g., beta = 0.057 in Fig. 4
Nearest-neighbour inductive coupling ratio Lg/L0 in eq. (8); no closed-form value is given for the device.
beta1 = fitted to FE mode frequencies in Fig. 4 and Fig. 9
Next-nearest-neighbour coupling introduced in Appendix B to improve the circuit model fit.
delta_p = fitted as sole parameter to FE crosstalk values in Fig. 6(c,d); also computed with simulated fp
The measured decay rate of the predicted K0 exponential is obtained by fitting to the same FE data in Fig. 6(c,d).
plasma frequency fp = fundamental mode from FE eigenmode simulation, 11.89 to 25.14 GHz for r = 0.05 to 0.4 mm (Table I)
Section IV uses the simulated fundamental frequency instead of eq. (4) when computing delta_p via eq. (15), so the analytical model is not tested independently for the realistic device.

assumptions (5)

domain assumption Wire-medium plasma frequency formula, eq. (4), from Refs. [17,19,20], accurately describes the shunt array in the homogenized limit.
Underpins the plasma model in Section II.B; used to derive eq. (6) and eq. (18).
domain assumption The finite-height, finite-size cavity modes with l = 0 and Ez polarization are governed by the same anisotropic plasma permittivity eq. (3) as an infinite wire-medium.
Section II.B extends the metamaterial result to a rectangular cavity with Lz << Lx, Ly.
domain assumption Mutual inductance coupling coefficient k can be set to unity in the circuit model, with differences absorbed into L0 and Lb.
Fig. 3 caption; this makes the 1D mapping and closed forms eq. (8) possible.
domain assumption Crosstalk is dominated by a single evanescent TM00 radial waveguide mode, and reflections from the receiving qubit are negligible.
Appendix C, eqs. (33)-(41); derives the K0 form of eqs. (14)-(16).
domain assumption The trans-impedance formula of Ref. [33], eq. (32), is valid for weakly anharmonic transmon qubits.
Used in Appendix C to convert the waveguide trans-impedance into J_ij.

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Pith. "Pith review of Modelling Enclosures for Large-Scale Superconducting Quantum Circuits." pith.science (2026). https://pith.science/paper/5IQWCSEG

@misc{pith2026190902104,
  author       = {Pith},
  title        = {Pith review of: Modelling Enclosures for Large-Scale Superconducting Quantum Circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5IQWCSEG}},
  note         = {Machine review of arXiv:1909.02104}
}

read the original abstract

Superconducting quantum circuits are typically housed in conducting enclosures in order to control their electromagnetic environment. As devices grow in physical size, the electromagnetic modes of the enclosure come down in frequency and can introduce unwanted long-range cross-talk between distant elements of the enclosed circuit. Incorporating arrays of inductive shunts such as through-substrate vias or machined pillars can suppress these effects by raising these mode frequencies. Here, we derive simple, accurate models for the modes of enclosures that incorporate such inductive-shunt arrays. We use these models to predict that cavity-mediated inter-qubit couplings and drive-line cross-talk are exponentially suppressed with distance for arbitrarily large quantum circuits housed in such enclosures, indicating the promise of this approach for quantum computing. We find good agreement with a finite-element simulation of an example device containing more than 400 qubits.

Figures

Figures reproduced from arXiv: 1909.02104 by the authors.

**Figure 2.** FIG. 2. Lowest 10 modes for a cavity containing a contain [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** FIG. 4. Lowest [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: FIG. 7. Cross-section of the inductively shunted cavity, now [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8. Circuit representation for the lowest [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9. Normalized relative error (NRE) Σ [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. Circuit model for two transmon qubits (with junc [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

36 extracted references · 34 canonical work pages

[1]

(11) where k2 = k2 x +k2 y, k2 0 = 1/(βa2), and kx and ky are deﬁned in eq. (10). In ﬁg. (4) we show results of HFSS eigenmode simula- tions of cavities containing inductive shunt arrays with r/a = 0.25. We ﬁnd good agreement to our circuit model, which improves further when we include a next-nearest- neighbour coupling parameter β1 (see Appendix B for de...

work page
[2]

Sheldon, L

S. Sheldon, L. S. Bishop, E. Magesan, S. Filipp, J. M. Chow, and J. M. Gambetta, Physical Review A 93, 012301 (2016)

work page 2016
[3]

Barends, C

R. Barends, C. Quintana, A. Petukhov, Y. Chen, D. Kafri, K. Kechedzhi, R. Collins, O. Naaman, S. Boixo, F. Arute, et al. , Physical Review Letters 123, 210501 (2019)

work page 2019
[4]

Heinsoo, C

J. Heinsoo, C. K. Andersen, A. Remm, S. Krinner, T. Walter, Y. Salath´ e, S. Gasparinetti, J.-C. Besse, A. Potoˇ cnik, A. Wallraﬀ,et al., Physical Review Applied 10, 034040 (2018)

work page 2018
[5]

M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Chow, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, Applied Physics Letters 96, 203110 (2010)

work page 2010
[6]

A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, et al. , Physical review let- ters 101, 080502 (2008)

work page 2008
[7]

T. G. McConkey, J. H. B´ ejanin, C. T. Earnest, C. R. H. McRae, Z. Pagel, J. R. Rinehart, and M. Mariantoni, Quantum Science and Technology 3, 034004 (2018)

work page 2018
[8]

Filipp, M

S. Filipp, M. G¨ oppl, J. M. Fink, M. Baur, R. Bianchetti, L. Steﬀen, and A. Wallraﬀ, Physical Review A 83, 063827 (2011)

work page 2011

Show all 36 references

[9]

H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair, G. Catelani, A. P. Sears, B. Johnson, M. J. Reagor, L. Frunzio, L. I. Glazman, et al., Physical Review Letters 107, 240501 (2011)

2011
[10]

N. T. Bronn, V. P. Adiga, S. B. Olivadese, X. Wu, J. M. Chow, and D. P. Pappas, Quantum science and technol- ogy 3, 024007 (2018)

2018
[11]

Wenner, M

J. Wenner, M. Neeley, R. C. Bialczak, M. Lenander, E. Lucero, A. D. O’Connell, D. Sank, H. Wang, M. Wei- des, A. N. Cleland, et al. , Superconductor Science and Technology 24, 065001 (2011)

2011
[12]

J. M. Gambetta, J. M. Chow, and M. Steﬀen, npj Quan- tum Information 3, 2 (2017)

2017
[13]

Vahidpour, W

M. Vahidpour, W. O’Brien, J. T. Whyland, J. Ange- les, J. Marshall, D. Scarabelli, G. Crossman, K. Ya- dav, Y. Mohan, C. Bui, et al. , arXiv preprint arXiv:1708.02226 (2017)

2017 arXiv
[14]

D.-R. W. Yost, M. E. Schwartz, J. Mallek, D. Rosenberg, C. Stull, J. L. Yoder, G. Calusine, M. Cook, R. Das, A. L. Day, et al. , arXiv preprint arXiv:1912.10942 (2019)

2019 arXiv
[15]

N. A. Nicorovici, R. C. McPhedran, and L. C. Botten, Physical Review E 52, 1135 (1995)

1995
[16]

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, and C. M. Soukoulis, Applied Physics Letters 65, 645 (1994)

1994
[17]

Preskill, Quantum 2, 79 (2018)

J. Preskill, Quantum 2, 79 (2018)

2018
[18]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Physical review letters 76, 4773 (1996)

1996
[19]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, Physical Review B 67, 113103 (2003)

2003
[20]

P. A. Belov, S. A. Tretyakov, and A. J. Viitanen, Jour- nal of electromagnetic waves and applications 16, 1153 (2002)

2002
[21]

Krynkin and P

A. Krynkin and P. McIver, Waves in Random and Com- plex Media 19, 347 (2009)

2009
[22]

C. E. Murray and D. W. Abraham, Applied Physics Let- ters 108, 084101 (2016)

2016
[23]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, Journal of Physics: Condensed Matter 10, 4785 (1998)

1998
[24]

Remski, Microwave Journal 43, 190 (2000)

R. Remski, Microwave Journal 43, 190 (2000)

2000
[25]

M. J. Hartmann, F. G. Brandao, and M. B. Plenio, Nature Physics 2, 849 (2006)

2006
[26]

D. E. Nagle, E. A. Knapp, and B. C. Knapp, Review of Scientiﬁc Instruments 38, 1583 (1967)

1967
[27]

T. P. Wangler, RF Linear accelerators (John Wiley & Sons, 2008)

2008
[28]

Marcuvitz, Waveguide handbook, 21 (Iet, 1951)

N. Marcuvitz, Waveguide handbook, 21 (Iet, 1951)

1951
[29]

Shi, Y.-H

T. Shi, Y.-H. Wu, A. Gonz´ alez-Tudela, and J. I. Cirac, Physical Review X 6, 021027 (2016)

2016
[30]

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, Nature Photonics 9, 326 (2015)

2015
[31]

Gonz´ alez-Tudela, C.-L

A. Gonz´ alez-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, Nature Photonics 9, 320 (2015)

2015
[32]

Rahamim, T

J. Rahamim, T. Behrle, M. J. Peterer, A. Patterson, P. A. Spring, T. Tsunoda, R. Manenti, G. Tancredi, and P. J. Leek, Applied Physics Letters 110, 222602 (2017)

2017
[33]

Patterson, J

A. Patterson, J. Rahamim, T. Tsunoda, P. Spring, S. Je- bari, K. Ratter, M. Mergenthaler, G. Tancredi, B. Vlas- takis, M. Esposito, et al. , Physical Review Applied 12, 064013 (2019)

2019
[34]

Solgun, D

F. Solgun, D. P. DiVincenzo, and J. M. Gambetta, IEEE Transactions on Microwave Theory and Techniques (2019)

2019
[35]

W. H. Hayt, J. E. Kemmerly, and S. M. Durbin, En- gineering circuit analysis , Vol. 214 (McGraw-Hill New York, 1978)

1978
[36]

Losonczi, Acta Mathematica Hungarica 60, 309 (1992)

L. Losonczi, Acta Mathematica Hungarica 60, 309 (1992)

1992

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

(11) where k2 = k2 x +k2 y, k2 0 = 1/(βa2), and kx and ky are deﬁned in eq. (10). In ﬁg. (4) we show results of HFSS eigenmode simula- tions of cavities containing inductive shunt arrays with r/a = 0.25. We ﬁnd good agreement to our circuit model, which improves further when we include a next-nearest- neighbour coupling parameter β1 (see Appendix B for de...

work page

[2] [2]

Sheldon, L

S. Sheldon, L. S. Bishop, E. Magesan, S. Filipp, J. M. Chow, and J. M. Gambetta, Physical Review A 93, 012301 (2016)

work page 2016

[3] [3]

Barends, C

R. Barends, C. Quintana, A. Petukhov, Y. Chen, D. Kafri, K. Kechedzhi, R. Collins, O. Naaman, S. Boixo, F. Arute, et al. , Physical Review Letters 123, 210501 (2019)

work page 2019

[4] [4]

Heinsoo, C

J. Heinsoo, C. K. Andersen, A. Remm, S. Krinner, T. Walter, Y. Salath´ e, S. Gasparinetti, J.-C. Besse, A. Potoˇ cnik, A. Wallraﬀ,et al., Physical Review Applied 10, 034040 (2018)

work page 2018

[5] [5]

M. D. Reed, B. R. Johnson, A. A. Houck, L. DiCarlo, J. M. Chow, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, Applied Physics Letters 96, 203110 (2010)

work page 2010

[6] [6]

A. A. Houck, J. A. Schreier, B. R. Johnson, J. M. Chow, J. Koch, J. M. Gambetta, D. I. Schuster, L. Frunzio, M. H. Devoret, S. M. Girvin, et al. , Physical review let- ters 101, 080502 (2008)

work page 2008

[7] [7]

T. G. McConkey, J. H. B´ ejanin, C. T. Earnest, C. R. H. McRae, Z. Pagel, J. R. Rinehart, and M. Mariantoni, Quantum Science and Technology 3, 034004 (2018)

work page 2018

[8] [8]

Filipp, M

S. Filipp, M. G¨ oppl, J. M. Fink, M. Baur, R. Bianchetti, L. Steﬀen, and A. Wallraﬀ, Physical Review A 83, 063827 (2011)

work page 2011