Crosstalk Dispersion and Spatial Scaling in Superconducting Qubit Arrays

TL;DR

The paper tackles the challenge of static crosstalk in large fixed-frequency superconducting qubit arrays by developing a unified framework that combines circuit-mediated exchange, enclosure-mediated (evanescent) coupling, and spectral detuning. It derives both a quantum effective Hamiltonian and a classical scaling theory, then validates them with a 4×4 transmon lattice featuring inductive shunt pillars that raise the enclosure cutoff and induce exponential spatial decay. ZZ couplings are calculated via perturbation theory and shown to be well captured by a scaling law that incorporates distance and detuning, aligning with experimental ZZ spectroscopy across NN and some NNN interactions. The work provides quantitative design principles for crosstalk mitigation—crucial for scaling quantum processors—by leveraging lattice geometry and tailored enclosure engineering to suppress correlated errors and enable more reliable multi-qubit operations.

Abstract

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 $4 \times 4$ 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.

Figures (11)

• Figure 1: Cross-sectional and three-dimensional schematic of a rectangular cavity enclosure containing transmon qubits $Q_{1}$ and $Q_{n}$, where $Q_{n}$ denotes an arbitrary qubit among the total of $N$ qubits. Two primary crosstalk pathways are illustrated: (i) cavity-mediated electromagnetic coupling through evanescent enclosure modes below cutoff (green arrow), characterized by coupling strengths $g_{c1}$, $g_{cn}$, and the cavity--cavity coupling $g_{cc}$; and (ii) direct circuit-mediated exchange interactions $g_{1n}$ arising from the inverse capacitance matrix $C^{-1}$. Each transmon displays its anharmonic energy-level structure---ground state $\lvert 0\rangle$, first excited state $\lvert 1\rangle$, and second excited state $\lvert 2\rangle$---along with the transition frequencies $\omega_{01}^{(1)}$ and $\omega_{12}^{(1)}$, etc. The colored standing-wave patterns represent electromagnetic enclosure modes that mediate long-range interactions, and $g_{1n}$ denotes the effective direct coupling between distant qubits $Q_{1}$ and $Q_{n}$. The parameter $\zeta_{1n}$ denotes the corresponding static ZZ interaction strength, defined as the state-dependent frequency shift $\zeta_{1n} = \omega_{1n}^{(11)} - \omega_{1n}^{(10)} - \omega_{1n}^{(01)} + \omega_{1n}^{(00)}$, which quantifies the correlated error induced by residual couplings.
• Figure 2: Equivalent circuit model representing qubits as linear LC resonators coupled through both direct capacitive coupling (capacitors $C$) and virtual processes mediated by intermediate electromagnetic modes through intermediate cavity modes with frequencies $\omega_{c_M}$ to $\omega_{c_1}$. The circuit elements represent the effective inductances and capacitances of the resonators-cavity system, where the intermediate modes act as harmonic oscillators that facilitate long-range coupling between spatially separated resonators.
• Figure 3: Simulated ZZ couplings in the $4 \times 4$ transmon lattice. The yellow markers shows the result from exact diagonalization of the model Hamiltonian in Eq. \ref{['eq:Hamiltonian_model_2']}, while the purple markers give the results of the scaling model fitted to the exact-diagonalization ZZ couplings. Note that the yellow markers saturate at $\sim \qty{e-8}{\kilo\hertz}$ beyond Manhattan distance 4. We attribute this saturation to the algorithmic precision of the exact-diagonalization routine.
• Figure 4: Device architecture and frequency allocation. (a) Optical micrograph of the fabricated $4\times4$ transmon lattice showing qubit pads (concentric light grey circles) connected by nearest-neighbor capacitive links, with inductive shunt pillar holes interleaved between qubit sites. Both arrays share the same square lattice pitch ($a = \qty{2}{\milli\metre}$). Inductive shunt pillars (Appendix \ref{['greenfunction']}) suppress enclosure-mediated coupling by raising the cavity cutoff above the qubit band. (b) Measured qubit frequencies form two interleaved bands at [range-phrase = --, range-units = single]4.84.9, with most detunings satisfying $|\omega_i-\omega_j| / 2 \pi < \langle \alpha/2 \pi \rangle_{16} = \qty{-196.4}{\mega\hertz}$ (straddling regime).
• Figure 5: Measured ZZ interactions and extracted $J$ couplings. (a) Measured ZZ interactions between nearest-neighbor qubit pairs were extracted via Ramsey spectroscopy by preparing each control qubit in either the ground or excited state and observing the resulting state-dependent frequency shift in the target qubit. To assess temporal frequency stability, repeated Ramsey measurements ($\sim 400$ per qubit) were performed over a 12-hour window, yielding an average standard deviation of 0.88 across the device --- below the smallest measured ZZ value reported in this work. (b) Estimated $J$ couplings by taking the first term in the analytical expression described in Eq. \ref{['eq:ZZ_12']} and applying it to the results in (a). Interaction maps of the measured ZZ interactions in (c) and calculated $J$ couplings in (d) are shown across both nearest-neighbor and longer-range qubit pairs in the lattice.