Spatial noise correlations beyond nearest neighbors in ${}^{28}\mathrm{Si}/$Si-Ge spin qubits

We detect correlations in qubit-energy fluctuations of non-neighboring qubits defined in isotopically purified Si/Si-Ge quantum dots. At low frequencies (where the noise is strongest), the correlation coefficient reaches 10% for a next-nearest-neighbor qubit-pair separated by 200 nm. Correlations with the charge-sensor signal reach up to 70%, proving that the observed noise is of electrical origin. A simple theoretical model quantitatively reproduces the measurements and predicts a polynomial decay of correlations with interqubit distance. Our results quantify long-range correlations of noise in quantum-dot spin-qubit arrays, essential for their scalability and fault tolerance.

Figure 1

(a) A false-color scanning electron microscope image of a device nominally identical to the one measured. The white scale bar indicates 100 nm. (b) The simulation of the $z$ component of the micromagnet field in the QD plane. (c) A schematic depiction of the structure layers, with the micromagnet (yellow) on top and a thin ${\mathrm{Si}\mathrm{O}}_2$ layer (red) right below the metallic gates.

Figure 2

(a) The power spectral density of the left (blue) and right (red) qubit-energy fluctuations. The dotted line corresponds to a $f^{-2}$ dependence as a reference. Inset: sample time traces of each qubit energy showing a two-level switching behavior; the color code follows that of the main panel. (b),(c) The correlation coefficient amplitude (b) and phase (c). In all panels, the color gradient represents the estimating distribution of the corresponding quantity at each frequency, with the maximum likelihood estimator marked as points (details are discussed in Ref. [61]); the continuous lines are calculated from the two-level system set given in Fig. 4.

Figure 3

Normalized cross-PSD between the qubit energies and the charge-sensor signals $V_L^{\uparrow ,\downarrow }$ and $V_R^{\uparrow ,\downarrow }$. The two panels show the correlations for (a) the left and (b) the right qubit. Similar to Fig. 2, the color gradients represent the probability distribution of the cross-PSD, with the maximum marked with circles. The corresponding phases and auto-PSDs are presented in Appendix pp2.

Figure 4

One possible set of TLSs that reproduces the experimental data. The tonality gives the switching time of the corresponding TLS. All TLSs are located 50 nm above the QD plane and have an opposite image charge 4 nm further above. The two disks in the center correspond to the QDs, with yellow, orange, and red representing three, two, and one standard deviations of the electron’s probability distribution, respectively, for a 2-meV parabolic confinement. The QDs are located at $(x,z)=(\pm 100\mathrm{nm},\pm 5\mathrm{nm})$. We used $m=0.2m_0$, with $m_0$ the free-electron mass and $ϵ=13$.

Figure 5

Average normalized correlation amplitude at 1 Hz as a function of the interqubit separation. Pink (gray) circles correspond to the simulation of $⟨r⟩$ with (without) screening due to the metallic gates. Shaded regions represent one standard deviation. Purple squares correspond to $\overline{r}$. For reference, the red dashed line corresponds to an exponential decay $\exp (-\alpha d)$, with $\alpha =0.0085{\mathrm{nm}}^{-1}$, while the magenta and black curves are proportional to $d^{-4.2}$ and $d^{-2}$, respectively. The purple line is $\propto d^{-5}$ following the result from Appendix pp4.

Figure 6

The main panel of Fig. 2 including the auto-power spectral density of the nuclear-spin noise (orange) calculated according to Eq. (A5) for $d=0$. Insets: cross power spectral density and correlation coefficient calculated for an interdot separation of 200 nm. The dotted lines are for reference.

Figure 7

Auto-PSDs of the charge-sensor voltages.

Figure 8

Phase of the cross-PSDs between the charge-sensor signals and the qubit energies.

Figure 9

Normalized cross-PSD amplitude calculated from an ensemble of 1000 sets of TLSs for 200 nm interqubit separation. The black lines correspond to the 1000 spectra of individual sets, and the ensemble average for $⟨r⟩$ is plotted in pink and for $\overline{r}$ in purple.

Figure 10

Average normalized cross-PSD amplitude $⟨r⟩$ at 1 Hz for different densities of TLSs. Vertical dashed lines give the mean distance between TLSs, equal to $1/\sqrt{2\pi n_c}$.

Figure 11

Another set of TLSs that fits the measured data.