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A Comparative Study of Coherent and Incoherent Error Models for superconducting Transmon Arrays

Terry Wonjun Park

Published Sep 9, 2026• 0 citations
Publisher

Abstract

Abstract Fabrication-induced variation in Josephson junction parameters ---shunt capacitance~$C$, normal-state resistance~$\RN$, and superconducting gap~$\Dgap$---introduces uncertainty in transmon qubit frequencies and leads to residual $ZZ$ crosstalk that limits multi-qubit gate fidelity. This work presents a quantitative, end-to-end simulation framework that maps junction-level disorder to GHZ state fidelity in a three-qubit fixed-coupling transmon array. Five concrete results are obtained. (1)~An analytic pipeline is constructed linking $(C,\RN,\Dgap)$ to qubit frequency $\omega_{01}$ and detuning $\Delta$, enabling direct propagation of fabrication noise into gate-level error rates. (2)~Monte Carlo analysis with 2\% parameter variation shows that normal-state resistance dominates detuning uncertainty, producing $\sigma_\Delta=0.113$\,GHz---approximately $1.4\times$ larger than variations from $C$ or $\Dgap$. (3)~Two error models are implemented and compared: a probabilistic depolarizing model and a coherent $ZZ$ model using the anharmonicity-corrected dispersive shift $\zeta_{ZZ}=g^2(1/\Delta-1/(\Delta+\alpha))$. Because Model 1 encodes only the \emph{magnitude} of the coupling-induced error while Model 2 encodes its \emph{signed, oscillatory phase}, the two models are not interchangeable estimates of the same quantity: their predictions diverge by up to $|\Delta F| \approx 0.66$ in GHZ fidelity near the boundary of the perturbation-validity exclusion zone, with Model 2 alone capturing the periodic revivals in fidelity that a magnitude-only stochastic model cannot represent by construction. (4)~The coherent model identifies the breakdown of second-order perturbation theory at $\Delta\approx|\alpha|=0.248$\,GHz, which, combined with fabrication-induced detuning spread, yields a conservative model-avoidance region $\Delta\in[0.02,0.47]$\,GHz within which non-perturbative treatment is required for quantitative characterization. (5)~Model-selection guidance is established: the probabilistic model provides a conservative bound for threshold estimation, while the coherent model is required for accurate hardware characterization and mitigation design. Finally, the framework is structured to allow systematic extension to more realistic noise channels, including leakage, correlated fabrication disorder, and non-perturbative multilevel dynamics, providing a pathway toward experimentally validated device-level error modeling.

DOI
10.21203/rs.3.rs-10911714/v1

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Abstract Fabrication-induced variation in Josephson junction parameters ---shunt capacitance~CCC, normal-state resistance~\RN\RN\RN, and superconducting gap~\Dgap\Dgap\Dgap---introduces uncertainty in transmon qubit frequencies and leads to residual ZZZZZZ crosstalk that limits multi-qubit gate fidelity. This work presents a quantitative, end-to-end simulation framework that maps junction-level disorder to GHZ state fidelity in a three-qubit fixed-coupling transmon array. Five concrete results are obtained. (1)~An analytic pipeline is constructed linking (C,\RN,\Dgap)(C,\RN,\Dgap)(C,\RN,\Dgap) to qubit frequency ω01\omega_{01}ω01​ and detuning Δ\DeltaΔ, enabling direct propagation of fabrication noise into gate-level error rates. (2)~Monte Carlo analysis with 2% parameter variation shows that normal-state resistance dominates detuning uncertainty, producing σΔ=0.113\sigma_\Delta=0.113σΔ​=0.113,GHz---approximately 1.4×1.4\times1.4× larger than variations from CCC or \Dgap\Dgap\Dgap. (3)~Two error models are implemented and compared: a probabilistic depolarizing model and a coherent ZZZZZZ model using the anharmonicity-corrected dispersive shift ζZZ=g2(1/Δ−1/(Δ+α))\zeta_{ZZ}=g^2(1/\Delta-1/(\Delta+\alpha))ζZZ​=g2(1/Δ−1/(Δ+α)). Because Model 1 encodes only the \emph{magnitude} of the coupling-induced error while Model 2 encodes its \emph{signed, oscillatory phase}, the two models are not interchangeable estimates of the same quantity: their predictions diverge by up to ∣ΔF∣≈0.66|\Delta F| \approx 0.66∣ΔF∣≈0.66 in GHZ fidelity near the boundary of the perturbation-validity exclusion zone, with Model 2 alone capturing the periodic revivals in fidelity that a magnitude-only stochastic model cannot represent by construction. (4)~The coherent model identifies the breakdown of second-order perturbation theory at Δ≈∣α∣=0.248\Delta\approx|\alpha|=0.248Δ≈∣α∣=0.248,GHz, which, combined with fabrication-induced detuning spread, yields a conservative model-avoidance region Δ∈[0.02,0.47]\Delta\in[0.02,0.47]Δ∈[0.02,0.47],GHz within which non-perturbative treatment is required for quantitative characterization. (5)~Model-selection guidance is established: the probabilistic model provides a conservative bound for threshold estimation, while the coherent model is required for accurate hardware characterization and mitigation design. Finally, the framework is structured to allow systematic extension to more realistic noise channels, including leakage, correlated fabrication disorder, and non-perturbative multilevel dynamics, providing a pathway toward experimentally validated device-level error modeling.
In the companion paper (Part I), we developed an optimal-control framework to suppress intrinsic and decoherence-induced errors in controlled-Z (CZ) gates between directly coupled, frequency-tunable transmons. Here, we extend this framework to incorporate uncertainties in coupling strength, anharmonicities, and zero-flux qubit frequencies, enabling the design of high-fidelity gates robust against noise and parameter uncertainties. We classify these uncertainties by whether they perturb the off-diagonal coupling strengths or the diagonal detunings of the Hamiltonian, and we identify distinct suppression mechanisms for each class. Coupling errors are mitigated through dynamical decoupling induced by strong detuning modulation. For a 30-ns gate, this strategy reduces the coupling error to 6.21×10−76.21\times10^{-7}6.21×10−7 and tolerates coupling-strength uncertainties up to 2π×42π\times42π×4 MHz while keeping total gate infidelity below 10−310^{-3}10−3. Detuning errors are minimized near a symmetric-detuning operating point that balances the two dominant leakage-mediated interactions. A 24-ns gate operated near this point tolerates individual uncertainties of up to 2π×52π\times52π×5 MHz in each relevant diagonal parameter under the same infidelity threshold. We further optimize gate performance under realistic conditions where intrinsic errors, decoherence, and parameter uncertainties coexist across six representative stages of progressively improving hardware performance. The resulting stage-specific controls achieve lower infidelity and greater robustness. Together with the companion study (Part I), this framework offers a practical approach to balance speed, fidelity, and robustness while reducing the need for exhaustive characterization and frequent recalibration in large-scale superconducting quantum processors.
Crosstalk between qubits fundamentally limits the scalability of quantum processors, necessitating physics-based models that can handle the complexity of large qubit arrays. Here, we develop a comprehensive theoretical and experimental framework that captures residual interactions between both adjacent and non-adjacent qubits in fixed-frequency transmon lattices. The model integrates the combined effects of exponential localization in banded capacitance matrices, suppression of virtual couplings through detuning products across intermediate modes, and evanescent decay of below-cutoff electromagnetic fields, yielding predictive scaling relations for coupling strength as a function of spatial separation and spectral detuning. Experimental characterization of a 4×44 \times 44×4 superconducting-qubit lattice with inductive shunt pillars reveals exponential spatial decay and frequency-dependent suppression consistent with theoretical predictions, achieving quantitative agreement for all nearest-neighbor couplings across the \qtyrange[range-phrase = --, range-units = single]{4}{6}{\giga\hertz} operating range. Our results show that standard dispersive Hamiltonian approximations systematically overestimate long-range coupling when spatial and spectral dependencies are neglected; these errors propagate into circuit simulation and design strategies. Our framework provides design guidance for crosstalk mitigation in larger-scale quantum processors under realistic fabrication constraints, addressing a bottleneck in scalability.
The reproducibility of qubit parameters is a challenge for scaling up superconducting quantum processors. Signal cross talk imposes constraints on the frequency separation between neighboring qubits. The frequency uncertainty of transmon qubits arising from the fabrication process is attributed to deviations in the Josephson junction area, tunnel barrier thickness, and the qubit shunt capacitor. We decrease the sensitivity to these variations by fabricating larger Josephson junctions and reduce the wafer-level standard deviation in resistance down to 2%. We characterize 32 identical transmon qubits and demonstrate the reproducibility of the qubit frequencies with a 40 MHz standard deviation (i.e., 1%) with qubit quality factors exceeding 2 million. We perform two-level-system (TLS) spectroscopy and observe no significant increase in the number of TLSs causing qubit relaxation. We further show by simulation that for our parametric-gate architecture, and accounting only for errors caused by the uncertainty of the qubit frequency, we can scale up to 100 qubits with an average of only three collisions between quantum-gate transition frequencies, assuming 22%2 cross talk and 99.999.9%99.9 target gate fidelity.
Fault-tolerant quantum computing requires quantum gates that combine short duration, high fidelity, and robustness to noise. As quantum processors scale, robustness becomes increasingly important because exhaustive characterization and frequent recalibration of individual gates become impractical. We present a comprehensive optimal-control framework for controlled-Z (CZ) gates between directly coupled, frequency-tunable transmons that systematically accounts for intrinsic Hamiltonian errors and realistic decoherence mechanisms, including energy relaxation and dephasing. Using smooth, single-shot flux pulses, we obtain 12-ns CZ gates---close to the theoretical speed limit of 11.6 ns---with intrinsic errors suppressed to the 2×10−52\times10^{-5}2×10−5 level. We construct dephasing noise models, calibrated against published experimental measurements, that capture fast dephasing from room-temperature control electronics and slow dephasing from 1/f1/f1/f flux noise. Using these models, we identify dephasing sweet spots that suppress sensitivity to dephasing noise. We further exploit transmon asymmetry as a design parameter to access distinct noise-resilient operating regimes. These results establish an experimentally applicable framework for balancing gate speed, fidelity, and noise robustness while reducing characterization and recalibration overhead.
This master thesis was dedicated to design, fabricate, and characterize subsystems of superconducting transmon qubits. Especially, focus was on investigating the predictability of the plasma frequency of the transmon, from which its operational properties are govern. In turn, the plasma frequency is governed by the normal state resistance of the two Josephson junctions along with its total capacitance. In\nprevious experiments, this frequency predictability relied entirely upon successful simulations of the system. However, even small simulation errors prove to generate\nsevere consequences for the system and a better, more systematic control is desired.\nIn order to identify and minimize this uncertainty, subsystems of the transmon were isolated and investigated. Test structures were fabricated and used to measure the normal state resistance of tunnel junctions with different areas and oxidation parameters using a four-point probe setup. A regime within which the reproducibility is increased was statistically identified. The shunt capacitance, i.e. the largest contribution to the total capacitance, of transmons was extracted by coupling identical capacitors to resonators. The coupling capacitance of a resonator is related to its external quality factor which was measured from analyzing the reflected microwave signal at cryogenic temperatures.\nThe smallest standard deviation of normal state resistances for 110 junction pairs was measured for junctions with an area 2.37 um^2 to be 1.48%, corresponding to an uncertainty contribution of 2 MHz for a designed plasma frequency of 6 GHz. Six resonators were measured throughout the shunt capacitance experiment and a linear t was used to compare the results with a written model based on conformal mapping.
Scalable superconducting quantum processors require balancing critical constraints in coherence, control complexity, and spectral crowding. Fixed-frequency architectures suppress flux noise and simplify control via all-microwave operations but remain limited by residual ZZ crosstalk. Here we propose a microwave-activated three-qubit gate protocol for fixed-frequency transmon qubits in the large-detuning regime (∣Δ∣≫g|Δ| \gg g∣Δ∣≫g), leveraging the third-order nonlinear interaction to coherently exchange ∣001⟩↔∣110⟩\ket{001} \leftrightarrow \ket{110}∣001⟩↔∣110⟩ states. By incorporating a phase-compensated optimization protocol, numerical simulations demonstrate a high average gate fidelity exceeding 99.9%99.9\%99.9%. Systematic error analysis identifies static long-range ZZ coupling as the dominant error source in multi-qubit systems, which can be suppressed via operations in the large-detuning regime (∼1\sim 1∼1 GHz). The protocol maintains process fidelities exceeding 98%98\%98% under decoherence, while demonstrating intrinsic robustness to fabrication-induced parameter variations and compatibility with existing all-microwave two-qubit gate architectures. This hardware-efficient strategy advances scalable quantum computing systems by improving coherence properties, reducing spectral congestion, and expanding the experimental toolkit for error-resilient quantum operations in the noisy intermediate-scale quantum era.