Mitigation of exchange cross-talk in dense quantum dot arrays

TL;DR

This work tackles the challenge of cross-talk in dense spin-qubit gate architectures by introducing a direct method to characterize barrier-to-barrier cross-talk on exchange interactions in a Ge/SiGe 2×4 quantum-dot array. The core approach tracks the singlet-triplet ($ST^-$) avoided crossing to extract a barrier cross-talk matrix, enabling a two-layer virtualization that orthogonally controls exchange couplings via virtual barrier gates. Validation in three four-spin chains demonstrates that the extracted cross-talk coefficients largely predict dynamics when nearest-neighbor exchanges are on, confirming the method’s applicability to longer spin chains and automated tuning. The results provide design guidance for scalable, multi-qubit quantum simulators and identify regimes where nonlinear cross-talk or leakage pose challenges, outlining paths toward even more robust cross-talk compensation in dense quantum-dot devices.

Abstract

Coupled spins in semiconductor quantum dots are a versatile platform for quantum computing and simulations of complex many-body phenomena. However, on the path of scale-up, cross-talk from densely packed electrodes poses a severe challenge. While cross-talk onto the dot potentials is nowadays routinely compensated for, cross-talk on the exchange interaction is much more difficult to tackle because it is not always directly measurable. Here we propose and implement a way of characterizing and compensating cross-talk on adjacent exchange interactions by following the singlet-triplet avoided crossing in Ge. We show that we can easily identify the barrier-to-barrier cross-talk element without knowledge of the particular exchange value in a 2x4 quantum dot array. We uncover striking differences among these cross-talk elements which can be linked to the geometry of the device and the barrier gate fan-out. We validate the methodology by tuning up four-spin Heisenberg chains. The same methodology should be applicable to longer chains of spins and to other semiconductor platforms in which mixing of the singlet and the lowest-energy triplet is present or can be engineered. Additionally, this procedure is well suited for automated tuning routines as we obtain a stand-out feature that can be easily tracked and directly returns the magnitude of the cross-talk.

Figures (25)

• Figure 1: (a) Schematics of the confinement potential for a chain of charges defined by the top gates in real voltage space (black solid line). A negative voltage pulse on the central barrier gate causes not only the middle tunnel barrier to be lowered but also shifts the electrochemical potentials of the nearby dots and the height of adjacent tunnel barriers (blue dashed line). (b) Commonly used virtual plunger gates work by applying a linear combination of gate voltages that keeps the electrochemical potentials of all other dots fixed. However, adjacent tunnel barrier heights are still affected and lateral shifts of charges are still present, although they might be slightly reduced. (c) If also the barriers are virtualized, a pulse on the middle barrier gate is compensated by suitable pulses on other barrier gates to keep the other tunnel barriers fixed and, ideally, counteract the lateral shifts of charges. In practice, however, only the combined effect of lateral shifts and tunnel barrier alterations can be compensated. A correct virtualization should allow orthogonal control of exchange interactions and enable a straightforward tuning of multi-spin chains. (d) Schematic of the 2$\times$4 dot array we use in this experiment. The dot plungers are labeled as $p_{i}$. Barriers $b_{ij}$ separate dots $i$ and $j$. The external magnetic field $B$ is applied in an in-plane direction marked by the arrow. (e) Energy level diagram of a two-spin system in a double quantum dot as a function of detuning $\epsilon$ (left) and tunnel coupling $t_c$ (right). The dashed circles mark the spin-orbit induced avoided crossings. At $|\epsilon|>U$ the two charges occupy the same dot ((2,0) and (0,2) charge regions). For $|\epsilon|<U$ the charges are shared between the two adjacent dots and the energy splittings are determined by the respective Zeeman energies and the exchange interaction. The position of the avoided crossing can be influenced by $\epsilon$ and $t_c$. (f) Measurement of the avoided crossing as a function of detuning and barrier voltage of $Q_{12}$, as described in the main text. The avoided crossing always occurs when $J=E_{T^-}$ constituting a constant-exchange feature which we are able to follow as a sharp reduction in singlet return probability $P_S^{12}$. At more positive values of $b_{12}'$, $S-T^0$ oscillations cause a reduced singlet return probability as well. As the barrier gets more negative, the exchange increases pushing the avoided crossing feature to smaller $\epsilon_{12}$. At $\epsilon_{12}=0$ all the exchange is induced by the barrier voltage $b_{12}'$.
• Figure 2: (a) Exchange oscillations in $Q_{56}$ as a function of $b'_{15}$ and dwell time $\tau$, as explained in the main text. (b) FFT of (a). We clearly see that the main frequency is reduced and for more negative values of $b'_{15}$ a second oscillation frequency appears. The frequency reduction is a sign of cross-talk, the appearance of a second frequency is due to a finite $J_{15}$. (c) Sketch of the experiment in (a). The orange circles depict the approximate charge positions when only $b'_{56}$ induces exchange. The blue dashed circles represent the shifted dot positions as we open $b'_{15}$. (d) Exchange oscillations in $Q_{56}$ as a function of $b'_{26}$ and dwell time $\tau$. (e) FFT of (d). We see a change in frequency of about $60\%$ over only $30mV$, indicating strong cross-talk. (f) Sketch of the experiment in (d) similar to (c). The fan-out of $b_{26}$ leads to a much larger cross-talk than for $b_{15}$ and may affect the position of all the nearby charges.
• Figure 3: (a) $ST^-$ avoided crossing of $Q_{56}$ as a function of $\mathrm{b'}_{56}$ on the horizontal axis and all the other barriers on the respective vertical axis. The plots are ordered to reflect the geometric location of the stepped gate. The position of the avoided crossing is reflected by a sharp decrease of the singlet return probability (see main text). From the fitted red dashed lines we extract the cross-talk elements $\alpha_{56}^{mn}$. The fact that we can fit all cross-talk features with a linear function confirms the assumption of linear barrier cross-talk. (b) $ST^-$ avoided crossing of $Q_{56}$ as a function of $\mathrm{b}^\dagger_{56}$ on the horizontal axis and all the other virtual barriers on their respective vertical axis. After the virtualization process the $ST^-$ avoided crossing position is only controlled by $b^\dagger_{56}$ as intended. To further verify that the exchange remains stable we plot exchange oscillations of $Q_{56}$ in the bottom left. We vary all the virtual barriers except $\mathrm{b}^\dagger_{56}$ together in the same range as in the individual plots. As desired, the exchange oscillations do not change in the ranges considered here. We repeat the same procedure on the other barrier gates (see Appendix section \ref{['sec:Cross-talk other barriers']}) and successively fill in the values for $\alpha_{ij}^{mn}$.
• Figure 4: (a) Summary of the exchange cross-talk elements $\alpha_{ij}^{mn}$ extracted for the barriers highlighted in bright green. We clearly see a reduction of cross-talk with distance as we expect for capacitive cross-talk. Positive (negative) cross-talk elements imply that an adjacent barrier gate enhances (reduces) a given exchange coupling. (b) Virtual gate matrix for the barrier gates showing the $\alpha_{ij}^{mn}$, summarizing the results from (a). Except for $\alpha_{26}^{mn}$, $\alpha_{37}^{mn}$, and $\alpha_{67}^{mn}$, all the elements $\alpha_{ij}^{mn}$ are measured. In the experiment the elements labeled 'n.m.' are replaced by 0. To be clear, this matrix reports how much each $b'_{ij}$ affects the various $b^\dagger_{mn}$. The barrier gate voltages $b'_{ij}$ needed to orthogonally control the respective exchange interactions via $b^\dagger_{mn}$ are obtained from the inverse of this matrix.
• Figure 5: (a) Resonant $\ket{S_{78}T^-_{34}}\leftrightarrow \ket{T^-_{78}S_{34}}$ condition as a function of $b^\dagger_{34}$ and $b^\dagger_{78}$ with an exchange $J_{48}\approx 2MHz$ induced by $b^\dagger_{48}$. We record the probability of measuring $\ket{S_{34}}$ after initializing $\ket{S_{78}T^-_{34}}$ and letting the system evolve for $\tau = 380ns$ corresponding to a near perfect inversion of population at the resonant condition marked by an increase in $P_{S}^{34}$. The red dots mark the theoretical resonant condition, based on the Zeeman energies and individual exchange dependencies, which agrees well with the data. (b) Resonant $\ket{S_{78}T^-_{34}}\leftrightarrow \ket{T^-_{78}S_{34}}$ oscillations as a function of dwell time $\tau$ and $b^\dagger_{34}-b^\dagger_{78}$. The barriers are scanned along the dashed line in (a). The maximum oscillation amplitude corresponds to the resonant condition and the frequency is $hf = J_{48}$. (c) Resonant $\ket{S_{78}T^-_{34}}\leftrightarrow \ket{T^-_{78}S_{34}}$ oscillations as a function of dwell time $\tau$ and $b^\dagger_{48}$ at the resonant condition. (d) FFT of (c). The red dashed line is the exchange dependence $J_{48}(b^\dagger_{48})$ extracted from the isolated $Q_{48}$ oscillations which matches well with the observed FFT peak. The latter yields $J_{48}(b^\dagger_{48})$ with the two neighbouring exchanges activated. The inset shows a sketch of the dots and interactions involved in the experiments (a)-(d). (e)-(h) Similar to (a)-(d) but for chain 2-1-5-6. In this case $b^\dagger_{15}$ was not virtualized due to the rather small cross-talk elements. (g) and (h) show data pertaining to the resonant $\ket{S_{12}T^0_{34}}\leftrightarrow \ket{T^0_{12}S_{34}}$ condition. The red dashed line in (h) is a fit and allows us to extract $J_{15}(b^\dagger_{15})$ even without complete virtualization. (i)-(l) Similar to (a)-(d) but for chain 1-2-3-4. While we find good agreement with the predicted resonant condition in (i), the oscillation frequency in (j) and (k) is different from the expected value extracted from the isolated $Q_{23}$ oscillations. We do find good agreement with the data if we correct the value $b^\dagger_{0,23}$ by $-6mV$. This suggests that some residual, possibly non-linear cross-talk remains, which will require more sophisticated mitigation strategies to account for.