Scaling of silicon spin qubits under correlated noise

Abstract

The path to fault-tolerant quantum computing hinges on hardware that scales while remaining compatible with quantum error correction (QEC). Silicon spin qubits are a leading hardware candidate because they combine industrial fabrication compatibility with a nanoscale footprint that could accommodate millions of qubits on a chip. However, their suitability for QEC remains uncertain since spatially correlated noise naturally emerges from the resulting close proximity of qubits. These correlations increase the likelihood of simultaneous errors and erode the redundancy that QEC depends on. Here we quantify the spatial extent of noise correlations in a five-qubit silicon array and assess their impact on QEC. We identify two distinct sources of correlated noise: global magnetic field drifts that generate perfectly correlated fluctuations, and charge noise from two-level fluctuators that produces short-range correlations decaying within neighboring qubits. While magnetic drifts represent a critical correlated noise source that can compromise QEC, they can be mitigated. In contrast, the measured charge noise correlations are moderate, electrically tunable, and compatible with fault-tolerant operation with minimal qubit overhead. Our results establish quantitative benchmarks for correlated noise and clarify how such correlations impact the viability of quantum error correction in scalable qubit arrays.

Figures (15)

• Figure 1: Noise correlation regimes and device layout. (a-c) Schematic illustrations of qubits subject to dephasing noise in different correlation regimes: (a) uncorrelated, (b) partially correlated, and (c) perfectly correlated. Qubits are represented as spheres with arrows accumulating random phases $\phi_i$, where the degree of correlation determines whether these phases are independent, partially shared, or identical. (d) Scanning electron microscope image of a device nominally identical to the one used in the experiment. Colored circles indicate the location of the five qubits, labeled Q1--Q5. Charge sensors $S_L$ and $S_R$ (white ovals) are used for spin-to-charge conversion and readout. An external in-plane magnetic field $\vec{B}_\mathrm{ext}$ is applied perpendicular to the qubit array. Global microwave control enables single-qubit rotations. (e) 3D rendering of the device structure. The $^{28}$Si quantum well hosting the qubits is shown in light blue. Charge noise arises from two-level fluctuators in the surrounding oxide layers and couples to the spins via the artificial spin-orbit field generated by a cobalt micromagnet (purple). (f) Measured individual Zeeman splittings for each qubit, showing the engineered energy gradient across the array. This linear gradient provides qubit addressability, allows the relative qubit positions to be inferred from their Zeeman energies, and sets the sensitivity to charge-noise-induced dephasing.
• Figure 2: Separation of global magnetic drift and local charge noise. (a) Measured time traces of the qubit energies over a 24-hour period, showing a clear negative drift attributed to a gradual loss of magnetization in the superconducting magnet. Lighter traces show the same data after subtracting a linear drift from each qubit individually. (b) Histograms of the energy fluctuations of qubit Q3 before (top) and after (bottom) drift removal. The drift causes a skewed distribution, which reverts to a Gaussian distribution after the removal. (c) Normalized cross-PSD between the most distant qubit pair, Q1--Q5. The upper panel shows the magnitude, and the lower panel the phase. Strong low-frequency correlations with phase around 0 are observed, consistent with a global magnetic field drift. After drift removal (faint traces), the magnitude drops and the phase is no longer locked at a single value, indicating uncorrelated noise. Occasional spikes of apparent correlation at low frequencies are attributed to the limitations of the simplistic linear drift removal and reduced statistical confidence in spectral estimates at these frequencies. A vertical dotted line marks a crossover, which takes place somewhere around $10^{-2}$ Hz, judged from the auto-PSDs in (d) transitioning from a uniform, drift-dominated regime to qubit-specific spectra with charge-noise features. (d) Auto-PSDs for all five qubits. Drift removal (faint colors) reduces the low-frequency noise by an order of magnitude. Above $10^{-2}$ Hz, the spectra become qubit-specific. Q3 exhibits a prominent Lorentzian peak, consistent with coupling to a TLF with a characteristic switching time of $0.7$ s, shown as a dotted-line fit. The dashed orange line shows the expected auto-PSD for a linear drift of 8 Hz/s, calculated using Eq. \ref{['eq:drift_psd']} (Methods). The solid gray line shows the predicted noise spectrum from nuclear spin diffusion, including oscillations of the electron wavefunction arising from the valley degree of freedom in silicon Rojas-Arias2026a. The measured spectra exceed the nuclear spin noise prediction, indicating that nuclear spin diffusion is not the dominant dephasing mechanism in our device, although a finite contribution from nuclear spins cannot be entirely excluded. The black dash-dotted line presents a reference $1/f$ power-law dependence.
• Figure 3: Spatial correlations and their tunability by gates. (a) Normalized cross-PSD magnitudes between Q2 and increasingly distant qubits: Q3 (top), Q4 (middle), and Q5 (bottom). A pronounced correlation peak is observed for Q2--Q3, which aligns with the Lorentzian feature in the auto-PSD of Q3 shown in Fig. \ref{['fig:auto-PSD']}, suggesting coupling to a shared TLF. The same feature appears in Q2--Q4 with reduced amplitude, and is no longer visible in Q2--Q5, showing a clear spatial decay of the correlation. The average correlation value for each pair is written in the upper left corner. (b) Average correlation matrices for the frequency range $[1\times10^{-2}~\mathrm{Hz},\ 4~\text{Hz}]$, extracted under two different offset values $\Delta V_B$ of the interdot barrier voltage between Q2 and Q3 (voltage conditions different from those in (a)). In each cell, the central number is the maximum of the Bayesian posterior, with top and bottom numbers indicating 90% confidence intervals. For positive $\Delta V_B$ (left), Q2 and Q3 are brought closer together, resulting in stronger correlations between them. For negative $\Delta V_B$ (right), Q2 and Q3 are pushed apart, their correlation weakens, while correlations Q1--Q2 and Q3--Q4 grow. The schematic (not to scale) qubit positions are illustrated above each matrix. The actual positional shifts can be estimated from the measured Zeeman energies and the known magnetic-field gradients (see Fig. \ref{['fig:device_f']}). Overall, it is clear that noise correlations can be changed by gate voltages.
• Figure 4: Scaling of noise correlations. Measured average correlations for all qubit pairs across multiple gate voltage configurations (colored symbols), evaluated after drift removal and over the frequency range $[1\times10^{-2}~\mathrm{Hz},\ 4~\text{Hz}]$, with circles and stars for the two cooldowns. White squares indicate the mean correlation for each neighbor order. The measured qubit energy detuning for each qubit pair is shown on the top horizontal axis and is converted to interqubit separation using the magnetic-field gradient along the qubit array axis of 0.051 mT/nm. The blue dashed line shows an exponential decay with correlation length $l_c=N_c L_q=81$ nm, corresponding to $N_c=0.75$ for an average nearest-neighbor qubit spacing of $L_q=108$ nm. The black curve shows the average correlation decay predicted by the TLF model with density $\rho_\text{TLF}=3\times10^{10}$ cm$^{-2}$, and the gray shaded region denotes the 10-90 percentile range across 1500 TLF ensembles, showing good agreement with the experimental data at short and intermediate separations. At larger interqubit distances, however, the measured correlations exhibit a statistically significant excess relative to the model, as analyzed in Extended Data Fig. \ref{['exfig:deviation']}, indicating residual long-range correlations beyond the TLF-only description. Our model has a single fit parameter, the density of TLFs. We show the sensitivity to this parameter by taking a density ten times lower than the best-fit value. The dotted line shows the predicted correlations, demonstrating that we can fit the TLF density with a resolution much better then an order-of-magnitude.
• Figure 5: QEC performance under correlated noise. (a) Logical error rate of the repetition code as a function of the code distance $d$ (equal to the number of physical qubits $N$ for this code) and correlation length $N_c$ for exponentially decaying correlations $c_{i,j} = \exp(-\left|\vec{r}_i-\vec{r}_j\right|/N_c L_q)$ (color gradient). Limiting cases of uncorrelated ($N_c \to 0$) and fully correlated ($N_c \to \infty$) noise are marked with colored circles; representative intermediate cases are shown with gray symbols. The solid black curve shows the logical error rate computed using the TLF model from Fig. \ref{['fig:scaling']}. (b) Upper panel: Comparison of repetition-code performance for 1D and 2D qubit layouts of up to 15 qubits. Circles (triangles) show the logical error rate for the 1D (2D) array. The limiting cases of uncorrelated and perfectly correlated noise are identical for both architectures, but the 2D case exhibits notably worse performance for partially correlated noise. Lower panel: Surface-code logical error rate for perfectly correlated (red) and uncorrelated (blue) noise. Diamonds with solid lines denote the lower bound of the calculation described in Methods. The shaded region spans the interval between this lower bound and a corresponding upper bound, with the true logical error rate lying within the band. For uncorrelated noise the band is essentially invisible, indicating tight bounds. While uncorrelated noise yields exponential suppression of logical errors with increasing distance, perfectly correlated noise results in a reduction of less than two orders of magnitude when increasing the distance from $d=1$ to $d=15$. Because the number of data qubits scales as $N=d^2$ (reaching 225 qubits at $d=15$), this limited suppression under perfectly correlated noise comes at substantial qubit overhead. The qualitative agreement with the repetition-code results in the upper panel indicates that the impact of correlated noise is not specific to a particular QEC architecture.