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1 DOI:.38/NNANO Bell s inequality violation with spins in silicon Juan P. Dehollain, Stephanie Simmons, Juha T. Muhonen, Rachpon Kalra, Arne Laucht, Fay Hudson, Kohei M. Itoh, David N. Jamieson, Jeffrey C. McCallum, Andrew S. Dzurak, and Andrea Morello STATISTICS, ERRORS AND FIDELITIES For the Bell s inequality violation experiment, each histogram in Figure 2 is constructed from Bell signal measurements. A Bell signal is calculated after extracting a set of 4 correlation values (E) corresponding to each axis projection combination. E is calculated from the averaged result of 3 pulse sequence repetitions and single-shot measurements. In total, each histogram is constructed from 2, single-shot measurements, which take hours to obtain. To complement the histograms in Figure 2 of the main text, Supplementary Fig. 2 shows further details on the E values extracted for each measurement. The number of single-shot repetitions per E measurement is chosen such that the error in the correlation value is dominated by the experimental imperfections and not by statistical error due to the probabilistic nature of the single-shot measurement. There is an additional limitation in the total amount of time available to perform the entire measurement, therefore a compromise needs to be reached between a number of single-shot measurements which minimizes the statistical errors and a number of Bell signals from which to construct an accurate histogram. For our chosen number of single-shot repetitions per E measurement (3), we expect a statistical error in E of σ E =2 (p( p))/n =.4 (where n is the number of single-shot measurements and p =.854 is the probability of obtaining E = after a perfect execution of the CHSH protocol). Supplementary Fig. 2 shows that our experiments returned, on average, errors ineof.6 (taken as the standard deviation of the data). This suggests that with 3 single-shots repetitions, we are in a regime where both statistical errors and experimental imperfections influence our observed error. It is likely that increasing the number of singleshot repetitions will narrow the width of the histogram and decrease our error, however this would involve significantly extending our total measurement time per Bell histogram (currently at hours), which is not feasible with our current experimental setup. The electron and nuclear gate fidelities (> 99%), as well as the electron and nuclear measurement fidelity ( 97% and > 99.9% respectively) have been previously documented in reference 8 of the main text. To extract the two-spin initialization fidelity, we perform two sets of 5 repetitions of the initialization sequence, followed by individual measurements of the nuclear spin for one set of repetitions, and electron spin (measured via the nucleus) for another set. We obtain P =.982(5) and P =.984(5) for the nuclear and electron measurement sets respectively. Each of these probabilities correspond to the populations of their respective two-spin manifolds (P = P + P and P = P + P ). If we assume P =, we can extract a two-spin initialization fidelity by intersecting the two measured populations F =(P P ) = 97()%. This leaves our total system performance primarily limited by the initialization and electron measurement fidelities. By mapping all the measurements onto the nucleus, we have been able to bypass the electron measurement fidelity bottleneck, leaving our system performance only limited mostly by the initialization fidelity. Even after the significant improvement in fidelity achieved in this work through selective initialization, this section of the qubit operation protocol is what limits both the Bell signals and the entangled state fidelity from the density matrix tomography measurement. This fact is clearly seen in our experiments, as the entangled state fidelity resulting from the density matrix tomography agrees with the initialization fidelity. Our current initialization fidelity of 97% is limited by a combination of effects, including the finite temperature (around mk, measured by monitoring the width of the SET coulomb peaks as a function of mixing chamber temperature) of the Fermi reservoir from which the initial spin-down electron is drawn, and spurious tunnelling events between the donor and the SET drain contact. This is by no means a fundamental limit, and we are continuously working to improve the filtering and thermalization of the system to further reduce the electron temperature. Finally, we comment on the asymmetry between the results obtained for the prepared entangled states ZQC and DQC. Although all the results for both states agree within their error margins, the results for DQC underperform those of ZQC for all experiments and measures. We have performed detailed numerical simulations of the entire initialization and measurement protocol to analyse this issue. We have found that, indeed, an asymmetry between ZQC and DQC appears, in agreement with experimental observations. The key to reproduce in the simulation the experimentally observed asymmetries in the S value measured for ZQC and DQC data was the combined inclusion of imperfect electron initialization, together with small errors modelled as either an instantaneous spin resonance frequency detuning, or errors in the NATURE NANOTECHNOLOGY 25 Macmillan Publishers Limited. All rights reserved

2 DOI:.38/NNANO 2 rotation angle of the control pulses. Conversely, introducing one individual type of error in the simulations did not give rise to asymmetries. We conclude that, overall, the asymmetric value of S for ZQC and DQC observed in the data is likely to be a genuine effect, which can be captured by numerical simulations. However, at this stage we have no clear insight into what precise combination of physical mechanisms causes the observed behaviour. This issue can be the topic of future studies. Supplementary Fig. Selective initialization protocol. Top: pulsing scheme used to initialize the two-spin system in the state, as described in the main text and Figure b-c. The SET current (blue traces) is tracked throughout the initialization sequence; high current implies ionized donor. The measurement is discarded if the donor is found to be ionized at the end of an electron initialization phase. Supplementary Fig. 2 Bell s inequality violation experiment details. a, Diagram showing how a π rf maps the parity observable onto the nuclear observable. The pulse coherently swaps the coefficients of the eigenstates, leaving the odd and even parity manifolds mapped on the and manifolds respectively. b, Correlation measurements used to construct the Bell signal histograms of Figure 2. Line colours correspond to each of the nuclear and electron qubit projection combinations, as shown in the legend (α and β are defined in Figure 2 of the main text). The top inset shows the cumulative moving average of P from a sample set of 3 4 single-shot measurements used to obtain one correlation set. For the QND measurements, we calculate the correlation E = 2P after each set of 3 single-shot measurements. 2 NATURE NANOTECHNOLOGY 25 Macmillan Publishers Limited. All rights reserved

3 DOI:.38/NNANO Supplementary Fig. 3 Density matrix tomography pulse sequences and fitting parameters. Left column: set of coherences measured by the tomography protocol. After preparing each tomography target state, a π rf and π mw2 pulse are applied with phase offsets ϕ and θ respectively. These pulses act as phase gates, which alter the phase of the coherence. The rest of the pulse sequence maps the coherence to be measured onto the manifold (column 2). A π/2 rf pulse is then applied to project the coherence onto the nuclear observable before performing a QND nuclear measurement. The nuclear spin proportion is plotted as we increment the phase offsets by ϕ and θ, and the resulting signals (see Supplementary Fig. 4) are fitted to P (n ph )=A sin(2πf p n ph + B)+C +.5. Here, n ph is the phase increment number and f p is a frequency in cycles/increment which is unique to each coherence (column 3). A and B are free fitting parameters from which we extract the off-diagonal element amplitude and phase respectively. The remaining free fitting parameter C is the offset of the measured signal, which gives information on the eigenstate populations (column 4). The diagonal elements of the density matrix are extracted by constructing a system of equations with all of the extracted offsets and =. The overdetermined system is solved using a non-negative least-squares solving algorithm. 3 Supplementary Fig. 4 Density matrix tomography detailed results. Plots correspond to the resulting signals obtained from following the density matrix tomography protocol for each of the coherences, after preparing the entangled states DQC and ZQC. The data (dots) is fitted (solid line) as described in Supplementary Fig. 3. The quadrature component is obtained by applying a π/2 phase offset to the final projective π/2 rf pulse and is fitted by adding π/2 to the sin argument in the fitting function. Note that some traces display oscillations at frequencies which are different from the characteristic frequency of the coherence. These oscillations arise from cross-talk between coherences that can result from the tomography pulse sequence. The cross-talk is minimized by carefully choosing the rate of phase increments for different coherences, to ensure that their Fourier transforms are well spaced from each other. Black dashed lines indicate offsets for the ideal states. The resulting density matrices for each entangled state are presented in numerical form at the bottom. We assume the matrix is Hermitian and apply a global phase correction so the coherence with greatest amplitude has zero phase. NATURE NANOTECHNOLOGY Macmillan Publishers Limited. All rights reserved

4 DOI:.38/NNANO SET SET V DG initialise initialise initialise ISET Time 4 NATURE NANOTECHNOLOGY 25 Macmillan Publishers Limited. All rights reserved

5 DOI:.38/NNANO a b Single-shot measurement repetitions ZQC (QND readout) DQC (QND readout).5.5.7(6).69(6).63(5) -.68(5) -.6(7).64(6) -.68(5) -.58(7) ZQC (Standard readout) DQC (Standard readout).5.73(5).57(6).52(6) -.69(6) (7).66(5) -.75(4) -.5(7) Correlation measurement repetitions (x3 single-shot measurements) NATURE NANOTECHNOLOGY Macmillan Publishers Limited. All rights reserved

6 DOI:.38/NNANO Manifold to map Phase-Map sequence Frequency (cycles/increment) Offset 6 NATURE NANOTECHNOLOGY 25 Macmillan Publishers Limited. All rights reserved

7 DOI:.38/NNANO Mapped manifold DQC Prepared State ZQC.75 In-phase Quadrature.5.25 In-phase Quadrature QND read proportion Phase increment index DQC ZQC NATURE NANOTECHNOLOGY Macmillan Publishers Limited. All rights reserved