Pith. sign in

REVIEW 3 major objections 6 minor 53 references

Purcell-Engineered Hybrid Coupler for Leakage-Suppressed Robust CZ Gates

T0 review · 3 major / 6 minor · reviewed 2026-07-30 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A hybrid coupler that pairs a transmon with a Purcell filter and notch resonator cuts superconducting CZ-gate leakage nearly twentyfold and raises worst-case fidelity above 99.6%.

desk verdict Solid coupler simulation with a real framing problem: the during-gate leakage win is mostly coherent, not dissipative. read the letter →

arxiv 2607.23463 v2 pith:334LCJ3O submitted 2026-07-26 quant-ph

Hui Wang , Feng Bao , Yan-Jun Zhao , Xun-Wei Xu This is my paper
classification quant-ph
keywords superconductingqubitsCZgatePurcellfilternotchresonatorleakagesuppressionengineereddissipationtransmoncouplerdressedstates
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

Superconducting CZ gates are limited not only by coherent control errors but by leakage into noncomputational states that can persist and spoil error correction. This paper argues that leakage can be fought with engineered dissipation, not only with pulse shaping and coupler design. It proposes a hybrid coupler: a nonlinear transmon plus a lossy Purcell-filter resonator and a weakly lossy notch resonator that together reshape both the coherent qubit–qubit interaction and the frequency-dependent environment. Dressed-eigenstate analysis and Lindblad simulations show leakage-like states decay about an order of magnitude faster than computational states, yielding F_avg = 99.74%, F_min = 99.62%, and maximum leakage of 1.6×10^{-3}—far better worst-case fidelity and roughly twentyfold less leakage than an optimized single-transmon coupler—while remaining stable across coherence sets and device-parameter scans. If the design holds in hardware, engineered dissipation becomes an extra design knob for robust two-qubit gates.

What carries the argument

The filter–notch subsystem: a lossy Purcell-filter mode coupled to a weakly dissipative notch resonator. The filter preferentially damps leakage-related dressed states (associated with 1→2 transitions); the notch reshapes the environmental response so computational 0→1 transitions stay protected. Engineered decay rates of dressed eigenstates plus Lindblad master-equation gate simulations quantify the selectivity and the resulting CZ performance.

What would settle it

Build the hybrid coupler and run a fixed-point CZ with the reported parameters (or a local re-optimization): if measured basis-state fidelities and leakage fail to beat an optimized single-transmon coupler by a large margin—especially if idle-to-gate flux/frequency ramps restore L_max near a few percent—the central claim does not hold in hardware.

Watch

Extended reading notes

Core claim

The authors claim that a Purcell-engineered notch-filter hybrid coupler—integrating a nonlinear transmon coupler with a coupled Purcell-filter and notch-resonator subsystem—combines coherent interaction engineering with leakage-selective dissipation and thereby substantially reduces residual leakage and improves worst-case computational-state fidelity for fixed-frequency CZ gates relative to an optimized single-transmon coupler. At the reported optimum they obtain F_avg = 99.74%, F_min = 99.62%, and L_max ≈ 1.6×10^{-3}, with the improvement persisting across coherence assumptions and broad filter/notch parameter ranges.

Load-bearing premise

The gate is treated as evolution under a fixed, time-independent Hamiltonian at one operating point; the real frequency-biasing ramp into and out of the gate, and any leakage-aware pulse shaping, are not simulated.

Editorial extensions

If this is right

  • Engineered dissipation becomes a usable design degree of freedom alongside coherent coupler engineering for superconducting CZ gates.
  • Maximum leakage can drop from ~3×10^{-2} to ~1.6×10^{-3} with F_min above 99.6% under the reported conditions.
  • The low-leakage, high-fidelity window stays open over broad filter and notch frequency/coupling ranges, easing fabrication tolerance.
  • Passive linear resonators (filter and notch) can supply this improvement without new active Josephson elements.

Reading between the lines

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

  • If leakage-like dressed states are preferentially emptied each gate, multi-cycle error-correction runs may see less correlated leakage propagation than single-gate L_bare alone suggests—something the paper flags as future work.
  • The same filter–notch spectral shaping could be tried on other leakage-prone two-qubit gates (e.g. iSWAP-family) where 1–2 transitions sit near environmental poles.
  • Combining this hardware environment with leakage-aware optimal control, rather than fixed-duration evolution, is a natural next optimization the baseline numbers leave open.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The authors propose a hybrid coupler for superconducting CZ gates in which a tunable transmon coupler is supplemented by a coupled Purcell-filter/notch-resonator subsystem. The filter branch adds a coherent exchange pathway (modifying the effective J_XX) while the filter–notch pair is designed to provide frequency-selective dissipation: strong engineered decay of leakage-like dressed states with suppressed decay of the computational manifold. Using full-Hamiltonian diagonalization (J_ZZ from Eq. 4; dressed decay rates from Eqs. 11–12) and Lindblad master-equation simulations in QuTiP, the authors compare the hybrid design against an optimized single-transmon coupler and a filter-only (g_fn=0) ablation, reporting F_avg=99.74%, F_min=99.62%, and L_max=1.6×10⁻³ at t_CZ=327.3 ns, with robustness scans over filter, notch, and qubit–filter parameters and six coherence sets.

Significance. If the results hold, this is a useful architecture-level contribution: leakage is a leading error source for superconducting QEC, and dissipation engineering for gates (as opposed to readout protection) is comparatively unexplored — the connection to passive leakage reset (Ref. 44, Thorbeck et al.) is timely. The study has real methodological strengths: explicit and separately optimized baselines (single coupler and filter-only ablation), coherence sweeps from conservative to near-lossless limits (Table III, Fig. 7), 2D robustness scans with J_ZZ re-extraction and local time re-optimization at each point (Figs. 9–11), and generally candid caveats (e.g., the acknowledgement that filter-induced relaxation converts leakage into computational errors rather than correcting them). These strengths make the numerical comparison credible as a simulation result. However, the paper is simulation-only, the fidelity metric is incomplete (below), and the central mechanistic framing — engineered dissipation as the operative leakage-suppression mechanism during the gate — is not quantitatively supported by the paper's own numbers.

major comments (3)
  1. [§III.A.1, Eqs. (11)–(12), Fig. 3, Table VI] The central framing — 'leakage-selective dissipation' as the mechanism suppressing CZ leakage (Abstract; Sec. II.C; Sec. III.A.1; Conclusion) — is quantitatively inconsistent with the paper's own numbers for the single-gate dynamics reported here. The reported average engineered rates are Γ_leak = 3.40×10⁻³ MHz (filter+notch) and 4.11×10⁻³ MHz (filter-only), i.e. ~3–4 kHz in ordinary-frequency units, corresponding to leakage-state lifetimes of order 50–300 µs. During t_CZ = 327.3 ns the probability of even one engineered decay event is at most Γ_leak × t_CZ ~ 10⁻³–10⁻², yet the headline result is a ~2.8×10⁻² absolute reduction of L_max. The suppression must therefore be dominated by coherent effects — the filter-mediated exchange J^(f)_XX (Eq. 7), dressed-state hybridization, and the retuned J_ZZ/t_CZ — i.e. multi-mode coherent coupler physics, with dissipation playing at most a minor wi
  2. [§III.B.1, definition of F_i and F_avg/F_min] The fidelity metric is a computational-basis population fidelity after local-Z correction: F_i is the final population in the target basis state. This metric is blind to any residual unitary error that is diagonal in the computational basis — in particular a conditional-phase error (gate angle ≠ π) or residual ZZ at the final time leaves all four basis-state populations unchanged and is invisible to F_avg, F_min, and to the time-sweep optimization that selected t_CZ. For a CZ gate, coherent conditional-phase error is one of the primary error channels, so the reported F_avg = 99.74% / F_min = 99.62% are upper bounds that exclude this channel; the comparison with the single-coupler baseline (Table V) inherits the same blindness. Please report a phase-sensitive figure of merit — e.g. average gate fidelity against the ideal CZ (computable from the simulated propagator on the computational su
  3. [§III opening paragraphs; §IV] All simulations use a time-independent Hamiltonian at a fixed operating point, and the idle-to-gate biasing transient is explicitly not modeled. Since the gate relies on a static J_ZZ ≈ 1/(2 t_CZ) ≈ 1.5 MHz, the architecture requires an idle configuration with strongly suppressed J_ZZ, and the ramp between the two will traverse the dressed spectrum whose selective damping is central to the mechanism. Nothing is said about the idle point: what is the residual J_ZZ there, and does the frequency trajectory cross filter/notch resonances that could either excite or beneficially reset leakage-like dressed states? At minimum, please report the idle-point parameters and residual J_ZZ, and discuss (or bound) the effect of a realistic flux ramp on the dressed-state selectivity that the fixed-point Lindblad runs rely on. This is a disclosed limitation, but for an architecture whose selling point is
minor comments (6)
  1. [Tables V–VI] The hybrid-coupler L_max differs between Table V (1.6×10⁻³) and Table VI (3.0×10⁻³); the text explains this as a common-evaluation vs. locally-optimized-point distinction, but the caption of Table VI should state explicitly which parameter set was used for each row, since readers will otherwise read the two tables as contradictory.
  2. [§III.B.2] The optimized single-transmon baseline parameters are never tabulated (only the hybrid parameters appear in Table I). Please provide them, and consider releasing the QuTiP scripts; both would substantially improve reproducibility.
  3. [§III.A.1] State explicitly whether the Γ values in Eqs. (11)–(12) and Fig. 3 are quoted as Γ/2π in MHz (ordinary-frequency units, as for J_ZZ) or as angular rates; the distinction matters by a factor of 2π for the lifetime estimates.
  4. [Fig. 2] Both panels are explicitly illustrative and 'do not correspond to the optimized device parameters'. It would be more informative to additionally show R(ω) for the actual Table I parameters, marking the dressed computational and leakage transition frequencies.
  5. [Fig. 7] The labels Low/Med./Good/Curr./Opt./Lossless should be mapped to Table III sets in the figure caption, not only in the text.
  6. [General] Typographical: 'achievesF avg' (Abstract); 'controlled-Z(CZ)'; inconsistent spacing around equations. Ref. [17] is an arXiv-only self-citation; update if published. The relation of the present work to Ref. [44] (bath engineering for passive leakage reset in transmons) deserves a fuller comparison, as that work is the closest prior art for the dissipative claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: fidelities and leakage are forward Lindblad outputs after parameter optimization, not identities forced by the inputs.

full rationale

This is a numerical device-design paper. The load-bearing chain is: (i) write a fixed-point multi-mode Hamiltonian including filter–notch modes, (ii) extract dressed J_ZZ / engineered decay rates by diagonalization, (iii) optimize coherent and filter parameters, (iv) evolve under the Lindblad master equation and report F_avg, F_min, L_max versus an optimized single-transmon baseline and ablations. None of these steps defines the reported metrics in terms of themselves, fits a parameter to a target fidelity/leakage and then “predicts” that same quantity, or imports a uniqueness theorem that forces the architecture. Self-citations [16,17] only supply standard virtual-exchange expressions for J^(c)_XX and J^(f)_XX (Eqs. 6–7); the leakage-suppression and fidelity claims are independently evaluated inside this manuscript via dressed-state decay rates (Eqs. 11–12, Fig. 3) and master-equation comparisons (Tables V–VI, Figs. 6–8). Skeptical concerns about whether kHz-scale Γ_leak can act during a 327 ns gate are correctness/mechanism issues, not circularity. Score 0; steps empty.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central numerical claim rests on standard circuit-QED and open-system axioms plus a large set of hand-chosen/optimized device parameters that define the operating point. No new fundamental physical constants are introduced; the main postulated object is the hybrid circuit itself. Headline fidelities are outcomes of multi-parameter optimization inside an idealized fixed-point Lindblad model, so free-parameter count is high even though circularity burden is low.

free parameters (4)
  • ω_f, ω_n, g_cf, g_fn, g_1f, g_2f, κ_f (and related Table I couplings/frequencies) = e.g. ω_f/2π=3.9378 GHz, ω_n/2π=4.5471 GHz, g_fn/2π=95.09 MHz, κ_f/2π=2.0 MHz (Table I–II)
    Jointly optimized (global then local refinement) to maximize computational fidelity and suppress leakage; reported F and L are at this operating point, not parameter-free predictions.
  • t_CZ (gate duration) = 327.3 ns
    Swept/locally refined around 1/(2|J_ZZ|) to maximize master-equation fidelity; optimal t_CZ=327.3 ns is chosen post-simulation.
  • Reference coherence set (T1,q, Tφ,q, T1,c, Tφ,c) = T1,q=Tφ,q=300 μs; T1,c=100 μs; Tφ,c=80 μs
    Primary results use long coherence (300 μs qubits) chosen to emphasize architecture vs ordinary relaxation; other sets only in sensitivity study.
  • g_12, g_1c, g_2c, ω_c, α_j, qubit frequencies = Table I (e.g. g_12/2π=9 MHz, ω_c/2π=5.675 GHz)
    Circuit parameters fixed/optimized as part of the coherent CZ design space that sets J_ZZ and residual J_XX.
assumptions (5)
  • domain assumption Markovian Lindblad evolution with independent collapse operators for qubit/coupler T1, Tφ and filter/notch loss fully captures gate errors of interest.
    Sec. III Eq. (9) and collapse-operator list; standard but excludes non-Markovian, 1/f flux noise during bias, and measurement backaction.
  • ad hoc to paper Gate operates under a time-independent Hamiltonian at fixed frequencies; idle–gate biasing transients can be ignored for architecture comparison.
    Stated explicitly at start of Sec. III; load-bearing for all reported F and L.
  • domain assumption Notch couples to qubits only via the filter (g_fn ≫ g_1n, g_2n), so direct notch–qubit channels are negligible.
    Eq. (1) and circuit-layout argument in Sec. II.A; justifies the five-mode Hamiltonian truncation of interactions.
  • domain assumption Circuit-QED capacitive coupling Hamiltonian (Eqs. 2–3) with given anharmonicities adequately represents the fabricated device.
    Sec. II.A; standard transmon/resonator model underlying all diagonalization and rates.
  • domain assumption Local-Z phase corrections after evolution are free and do not hide leakage or non-computational population.
    Sec. III.B; standard virtual-Z accounting, but fidelity definitions depend on it.
invented entities (1)
  • Purcell-engineered notch-filter hybrid coupler (transmon coupler + lossy Purcell filter F + auxiliary notch N as one subsystem)
    purpose: Provide simultaneous coherent pathway reshaping and leakage-selective dissipation for CZ gates.
    The integrated filter–notch-plus-nonlinear-coupler arrangement is the paper’s proposed object; performance claims are about this entity under simulation, with no fabricated-device evidence in the manuscript.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Purcell-Engineered Hybrid Coupler for Leakage-Suppressed Robust CZ Gates." pith.science (2026). https://pith.science/paper/334LCJ3O

@misc{pith2026260723463,
  author       = {Pith},
  title        = {Pith review of: Purcell-Engineered Hybrid Coupler for Leakage-Suppressed Robust CZ Gates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/334LCJ3O}},
  note         = {Machine review of arXiv:2607.23463}
}
abstract

We propose a Purcell-engineered notch-filter hybrid coupler for superconducting controlled-$Z$ (CZ) gates that combines coherent interaction engineering with leakage-selective dissipation. The architecture integrates a nonlinear transmon coupler with a coupled Purcell-filter and notch-resonator subsystem, providing additional control over both the coherent interaction pathways and the engineered dissipative environment. The filter branch reshapes the effective interaction pathways, while the notch resonator further tailors the frequency response of the coupled filter network and preserves strong leakage-selective dissipation. Using dressed-eigenstate analysis together with Lindblad master-equation simulations, we show that the proposed architecture substantially reduces leakage and improves the worst-case computational-state fidelity compared with an optimized single-transmon coupler while remaining robust over a broad range of coherence assumptions and device parameters. The optimized gate achieves $F_{\rm avg}=99.74\%$, $F_{\rm min}=99.62\%$, and a maximum leakage probability of $1.6\times10^{-3}$. These results demonstrate that engineered dissipation complements conventional coherent interaction engineering and provides an additional design degree of freedom for realizing robust, high-fidelity superconducting CZ gates.

Figures

Figures reproduced from arXiv: 2607.23463 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the Purcell-engineered filter–notch hy [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Representative dependence of the effective trans [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Engineered decay rates for representative computational and leakage-related eigenstates. Results are shown for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Random-state bare-projector leakage as a func [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Final local optimization of the hybrid coupler. [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Structure comparison between the optimized [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison between the optimized hybrid coupler [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Robustness of the optimized hybrid coupler against [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Robustness of the full filter–notch hybrid coupler against variations of the notch frequency [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Robustness of the optimized hybrid coupler against [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

53 extracted references · 1 canonical work pages

  1. [1]

    Mechanism of Leakage Suppression We investigate the leakage-suppression mechanism by analyzing the engineered decay rates of dressed eigen- states. Figure 2(b) provides a qualitative frequency- domain picture in which computational transitions are weakly coupled to the environment, whereas leakage- related transitions experience stronger effective dissipa...

  2. [2]

    III A explains the microscopic origin of leakage suppression

    Random-State Bare-Projector Leakage The dressed-state analysis in Sec. III A explains the microscopic origin of leakage suppression. To evaluate the resulting CZ-gate performance, however, leakage is quantified by the population outside the bare computa- tional subspace. To characterize leakage beyond the four computational basis states, we consider an en...

  3. [3]

    The corresponding CZ-gate duration is estimated ast CZ ≈ 1/(2|JZZ |)

    Final Local Optimization of the Hybrid Coupler The optimization simultaneously seeks to (i) maxi- mize the conditional interaction strength|J ZZ |, (ii) sup- press the residual transverse couplingJ XX , and (iii) re- duce residual leakage through the combined effects of co- herent state hybridization and frequency-selective cou- pling to the engineered fi...

  4. [4]

    Comparison with an Optimized Single-Transmon Coupler To evaluate the proposed architecture, we compare the optimized hybrid coupler with an optimized single- transmon coupler across the coherence sets listed in Ta- ble III. The qubit and coupler relaxation and dephasing times are varied from conservative experimental values to a near-lossless reference, a...

  5. [5]

    This comparison iso- lates the respective contributions of the filter branch and the notch resonator to leakage suppression

    Role of the Filter Branch and Notch Resonator To distinguish the roles of the different circuit compo- nents, we compare three architectures using the same master-equation evaluation procedure: an optimized single-transmon coupler, a hybrid coupler with the filter branch but without the notch resonator (g f n= 0), and the full filter–notch hybrid coupler....

  6. [6]

    Unlike the auxiliary notch resonator, the filter couples directly to the nonlinear transmon coupler throughg cf and indirectly to the qubits through the weak qubit–filter couplings

    Robustness Against Filter Parameters We first examine the robustness with respect to the primary filter parameters, namely the Purcell-filter fre- quencyω f and the coupler–filter coupling strengthg cf . Unlike the auxiliary notch resonator, the filter couples directly to the nonlinear transmon coupler throughg cf and indirectly to the qubits through the ...

  7. [7]

    Robustness Against Notch Parameters We next investigate the robustness with respect to the auxiliary notch parameters, namely the notch frequency ωn and the filter–notch couplingg f n. Because the notch resonator couples only to the Purcell filter, these parame- ters primarily reshape the frequency response of the cou- pled filter–notch subsystem while on...

  8. [8]

    To evaluate this sensitivity, we perform a fine asymmetric scan ofg 1f andg 2f around the optimized operating point, while keeping all other coherent parameters fixed

    Robustness Against Qubit–Filter Coupling Variations A practical implementation also requires robustness against fabrication-induced variations in the weak direct qubit–filter couplings. To evaluate this sensitivity, we perform a fine asymmetric scan ofg 1f andg 2f around the optimized operating point, while keeping all other coherent parameters fixed. Fig...

Show all 53 references
  1. [9]

    Acharyaet al.(Google Quantum AI and Collabora- tors), Quantum error correction below the surface code threshold, Nature638, 920–926 (2025)

    R. Acharyaet al.(Google Quantum AI and Collabora- tors), Quantum error correction below the surface code threshold, Nature638, 920–926 (2025)

    2025

  2. [10]

    Acharyaet al.(Google Quantum AI and Collabora- tors), Suppressing quantum errors by scaling a surface code logical qubit, Nature614, 676–681 (2023)

    R. Acharyaet al.(Google Quantum AI and Collabora- tors), Suppressing quantum errors by scaling a surface code logical qubit, Nature614, 676–681 (2023)

    2023

  3. [11]

    Blais, R.-S

    A. Blais, R.-S. Huang, A. Wallraff, S.M. Girvin, and R.J. Schoelkopf, Cavity quantum electrodynamics for super- conducting electrical circuits: An architecture for quan- tum computation, Phys. Rev. A69, 062320 (2004)

    2004

  4. [12]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buellet al., Quantum supremacy using a pro- grammable superconducting processor, Nature574, 505– 510 (2019)

    2019

  5. [13]

    Koch, T.M

    J. Koch, T.M. Yu, J. Gambetta, A.A. Houck, D.I. Schus- ter, J. Majer, A. Blais, M.H. Devoret, S.M. Girvin, and R.J. Schoelkopf, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A76, 042319 (2007)

    2007

  6. [14]

    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, Appl. Phys. Rev.6, 021318 (2019)

    2019

  7. [15]

    F. Yan, P. Krantz, Y. Sung, M. Kjaergaard, D.L. Camp- bell, J.I.-J. Wang, T.P. Orlando, S. Gustavsson, and W.D. Oliver, Tunable coupling scheme for implement- ing high-fidelity two-qubit gates, Phys. Rev. Applied10, 054062 (2018). 13

    2018

  8. [16]

    Y. Sung, L. Ding, J. Braum¨ uller, A. Veps¨ al¨ ainen, B. Kannan, M. Kjaergaard, A. Greene, G.O. Samach, C. McNally, D. Kim, A. Melville, B.M. Niedzielski, M.E. Schwartz, J.L. Yoder, T.P. Orlando, S. Gustavsson, L. Viola, and W.D. Oliver, Realization of high-fidelity CZ and ZZ...

    2021

  9. [17]

    Barends, J

    R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Camp- bellet al., Superconducting quantum circuits at the sur- face code threshold for fault tolerance, Nature508, 500– 503 (2014)

    2014

  10. [18]

    Li, J.-C

    T.-M. Li, J.-C. Zhang, B.-J. Chen, K. Huang, H.-T. Liu, Y.-X. Xiao, C.-L. Deng, G.-H. Liang, C.-T. Chen, Y. Liu et al., High-precision pulse calibration of tunable cou- plers for high-fidelity two-qubit gates in superconduct- ing quantum processors, Phys. Rev. Applied23, 024059 (2025)

    2025

  11. [19]

    Y. Xu, J. Chu, J. Yuan, J. Qiu, Y. Zhou, L. Zhang, X. Tan, Y. Yu, S. Liu, J. Li, F. Yan, and D. Yu, High-fidelity, high-scalability two-qubit gate scheme for superconduct- ing qubits, Phys. Rev. Lett.125, 240503 (2020)

    2020

  12. [20]

    Goto, Double-transmon coupler: Fast two-qubit gate with no residual coupling for highly detuned supercon- ducting qubits, Phys

    H. Goto, Double-transmon coupler: Fast two-qubit gate with no residual coupling for highly detuned supercon- ducting qubits, Phys. Rev. Applied18, 034038 (2022)

    2022

  13. [21]

    R. Li, K. Kubo, Y. Ho, Z. Yan, Y. Nakamura, and H. Goto, Realization of high-fidelity CZ gate based on a double-transmon coupler, Phys. Rev. X14, 041050 (2024)

    2024

  14. [22]

    D. L. Campbell, A. Kamal, L. Ranzani, M. Senatore, and M. D. LaHaye, Modular tunable coupler for supercon- ducting circuits, Phys. Rev. Applied19, 064043 (2023)

    2023

  15. [23]

    Y. Chen, C. Neill, P. Roushan, N. Leung, M. Fang, R. Barends, J. Kelly, B. Campbell, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, A. Megrant, J.Y. Mutus, P.J.J. O’Malley, C.M. Quintana, D. Sank, A. Vainsencher, J. Wenner, T.C. White, M.R. Geller, A.N. Cleland, and J.M. Martin...

    2014

  16. [24]

    Wang, Y.-J

    H. Wang, Y.-J. Zhao, H.-C. Sun, X.-W. Xu, Y. Li, Y. Zheng, Q. Liu, and R. Li, Controlling the qubit-qubit coupling in the superconducting circuit with double- resonator couplers, Phys. Rev. A109, 012601 (2024)

    2024

  17. [25]

    Tsai, Control the qubit-qubit cou- pling with double superconducting resonators, https://doi.org/10.48550/arXiv.2602.11576 (2026)

    Hui Wang, Rui Wang, Daichi Sugiyama, Chih- Yao Shih, Ching-Yeh Chen, Hiroto Mukai, Hang Xue, J.S. Tsai, Control the qubit-qubit cou- pling with double superconducting resonators, https://doi.org/10.48550/arXiv.2602.11576 (2026)

    2026 doi arXiv

  18. [26]

    Motzoi, J.M

    F. Motzoi, J.M. Gambetta, P. Rebentrost, and F.K. Wil- helm, Simple pulses for elimination of leakage in weakly nonlinear qubits, Phys. Rev. Lett.103, 110501 (2009)

    2009

  19. [27]

    Gambetta, F

    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, Phys. Rev. A 83, 012308 (2011)

    2011

  20. [28]

    Ghosh, A

    J. Ghosh, A. G. Fowler, J. M. Martinis, and M. R. Geller, Understanding the effects of leakage in superconduct- ing quantum error detection circuits, Phys. Rev. A 88, 062329 (2013)

    2013

  21. [29]

    Ghosh and A

    J. Ghosh and A. G. Fowler, A leakage-resilient approach to fault-tolerant quantum computing with superconduct- ing elements, Phys. Rev. A 91, 020302(R) (2015)

    2015

  22. [30]

    Suchara, A

    M. Suchara, A. W. Cross, and J. M. Gambetta, Leakage suppression in the toric code, Quantum Inf. Comput. 15, 997–1016 (2015)

    2015

  23. [31]

    McEwen, D

    M. McEwen, D. Kafri, Z. Chen, J. Atalaya, K. J. Satzinger, C. Quintana, P. V. Klimov, D. Sank, C. Gid- ney, A. G. Fowler, F. Arute, K. Arya, B. Buckley, B. Burkett, N. Bushnell, B. Chiaro, R. Collins, S. Demura, A. Dunsworth, C. Erickson, B. Foxen, M. Giustina, T. Huang, S. Ho...

    2021

  24. [32]

    C. J. Wood and J. M. Gambetta, Quantification and characterization of leakage errors, Phys. Rev. A 97, 032306 (2018)

    2018

  25. [33]

    X. Yang, J. Chu, Z. Guo, W. Huang, Y. Liang, J. Liu, J. Qiu, X. Sun, Z. Tao, J. Zhanget al., Coupler- assisted leakage reduction for scalable quantum error cor- rection with superconducting qubits, Phys. Rev. Lett. 133, 170601 (2024)

    2024

  26. [34]

    K. W. Murch, U. Vool, D. Zhou, S. J. Weber, S. M. Girvin, and I. Siddiqi, Cavity-Assisted Quantum Bath Engineering, Phys. Rev. Lett. 109, 183602 (2012)

    2012

  27. [35]

    Shankar, M

    S. Shankar, M. Hatridge, Z. Leghtas, K. M. Sliwa, A. Narla, U. Vool, S. M. Girvin, L. Frunzio, M. Mirrahimi, and M. H. Devoret, Autonomously Stabilized Entangle- ment between Two Superconducting Quantum Bits, Na- ture 504, 419–422 (2013)

    2013

  28. [36]

    Leghtas, U

    Z. Leghtas, U. Vool, S. Shankar, M. Hatridge, S. M. Girvin, M. H. Devoret, and M. Mirrahimi, Stabilizing a Bell State of Two Superconducting Qubits by Dissipa- tion Engineering, Phys. Rev. A 88, 023849 (2013)

    2013

  29. [37]

    Purcell, Spontaneous emission probabilities at radio frequencies, Phys

    E.M. Purcell, Spontaneous emission probabilities at radio frequencies, Phys. Rev.69, 681 (1946)

    1946

  30. [38]

    Reed, B.R

    M.D. Reed, B.R. Johnson, A.A. Houck, L. DiCarlo, J.M. Chow, D.I. Schuster, L. Frunzio, and R.J. Schoelkopf, Fast reset and suppressing spontaneous emission of a superconducting qubit, Appl. Phys. Lett.96, 203110 (2010)

    2010

  31. [39]

    Jeffrey, D

    E. Jeffrey, D. Sank, J.Y. Mutus, T.C. White, J. Kelly, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Megrant, P.J.J. O’Malley, C. Neill, P. Roushan, A. Vainsencher, J. Wenner, A.N. Cleland, and J.M. Marti- nis, Fast accurate state measurement with superconduct- ing...

    2014

  32. [40]

    Bronn, Y

    N.T. Bronn, Y. Liu, J.B. Hertzberg, A.D. C´ orcoles, A.A. Houck, J.M. Chow, and J.M. Gambetta, Broadband fil- ters for abatement of spontaneous emission in circuit quantum electrodynamics, Appl. Phys. Lett.107, 172601 (2015)

    2015

  33. [41]

    Quan- tum theory of a bandpass Purcell filter for qubit read- out,

    E. A. Sete, J. M. Martinis, and A. N. Korotkov, “Quan- tum theory of a bandpass Purcell filter for qubit read- out,” Phys. Rev. A 92, 012325 (2015)

    2015

  34. [42]

    High-fidelity readout in circuit quantum electrodynamics using the Jaynes-Cummings nonlinearity,

    M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I. Schuster, L. Frunzio, and R. J. Schoelkopf, “High-fidelity readout in circuit quantum electrodynamics using the Jaynes-Cummings nonlinearity,” Phys. Rev. Lett. 105, 173601 (2010). 14

    2010

  35. [43]

    N. T. Bronn, E. Magesan, N. A. Masluk, J. M. Chow, J. M. Gambetta, and M. Steffen, Reducing Sponta- neous Emission in Circuit Quantum Electrodynamics by a Combined Readout/Filter Technique, Phys. Rev. A 91, 062318 (2015)

    2015

  36. [44]

    Blais, A

    A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff, Circuit quantum electrodynamics, Rev. Mod. Phys. 93, 025005 (2021)

    2021

  37. [45]

    Gong, et al., Quantum walks on a programmable twodimensional 62-qubit superconducting processor, Sci- ence 372, 948 (2021)

    M. Gong, et al., Quantum walks on a programmable twodimensional 62-qubit superconducting processor, Sci- ence 372, 948 (2021)

    2021

  38. [46]

    Hui Wang, Yan-Jun Zhao, Rui Wang, Xun-Wei Xu, Qiang Liu, Jianhua Wang, and Changxin Jin, Frequency Adjustable Resonator as a Tunable Coupler for Xmon Qubits, J. Phys. Soc.Jpn. 91, 104005 (2022)

    2022

  39. [47]

    C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differen- tial equations and the master equation, Phys. Rev. A 31, 3761 (1985)

    1985

  40. [48]

    On the generators of quantum dynamical semigroups,

    G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys. 48, 119–130 (1976)

    1976

  41. [49]

    Completely positive dynamical semigroups of N-level systems,

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” J. Math. Phys. 17, 821–825 (1976)

    1976

  42. [50]

    J. R. Johansson, P. D. Nation, and F. Nori: ”QuTiP 2: A Python framework for the dynamics of open quantum systems.”, Comput. Phys. Commun. 184, 1234 (2013)

    2013

  43. [51]

    J. R. Johansson, P. D. Nation, and F. Nori: ”QuTiP: An open-source Python framework for the dynamics of open quantum systems.”, Comput. Phys. Commun. 183, 1760 (2012)

    2012

  44. [52]

    High-fidelity gates in a transmon using bath engineering for passive leakage reset,

    T. Thorbeck, A. McDonald, O. Lanes, J. Blair, G. Keefe, A. A. Stabile, B. Royer, L. C. G. Govia, and A. Blais, “High-fidelity gates in a transmon using bath engineering for passive leakage reset,” arXiv:2411.04101

    arXiv

  45. [53]

    D. C. McKay, S. Sheldon, J. M. Chow, and J. M. Gam- betta, Efficient Z Gates for Quantum Computing, Phys. Rev. A 96, 022330 (2017)

    2017

Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.