A silicon quantum-dot-coupled nuclear spin qubit

A silicon quantum-dot-coupled nuclear spin qubit

Bas Hensen    Wister Huang    Chih-Hwan Yang    Kok Wai Chan    Jun Yoneda1    Tuomo Tanttu    Fay E. Hudson    Arne Laucht Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, New South Wales 2052, Australia    Kohei M. Itoh School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan    Andrea Morello    Andrew S. Dzurak b.hensen@unsw.edu.au,wister.huang@unsw.edu.au, a.dzurak@unsw.edu.au Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, The University of New South Wales, Sydney, New South Wales 2052, Australia
thanks: These authors contributed equallythanks: These authors contributed equally

Nuclear spins in the solid state have long been envisaged as a platform for quantum computingKane1998silicon-based (); Ladd2002All-Silicon (); Skinner2003Hydrogenic (), due to their long coherence timesMaurer2012Room-Temperature (); Saeedi2013Room-Temperature (); Muhonen2014Storing () and excellent controllabilityVandersypen2005NMR (). Measurements can be performed via localised electrons, for example those in single atom dopantsPla2012single-atom (); Pla2013High-fidelity (); Pla2014Coherent () or crystal defectsChildress2006Coherent (); Neumann2010Single-Shot (); Abobeih2018One-second (). However, establishing long-range interactions between multiple dopants or defects is challengingBernien2013Heralded (); Tosi2017Silicon (); Humphreys2018Deterministic (). Conversely, in lithographically-defined quantum dots, tuneable interdot electron tunnelling allows direct coupling of electron spin-based qubits in neighbouring dotsNowack2011Single-Shot (); Veldhorst2014addressable (); Veldhorst2015two-qubit (); Watson2018programmable (); Zajac2018Resonantly (); Huang2018Fidelity (). Moreover, compatibility with semiconductor fabrication techniquesMaurand2016CMOS () provides a compelling route to scaling to large numbers of qubits. Unfortunately, hyperfine interactions are typically too weak to address single nuclei. Here we show that for electrons in silicon metal–oxide–semiconductor quantum dots the hyperfine interaction is sufficient to initialise, read-out and control single silicon-29 nuclear spins, yielding a combination of the long coherence times of nuclear spins with the flexibility and scalability of quantum dot systems. We demonstrate high-fidelity projective readout and control of the nuclear spin qubit, as well as entanglement between the nuclear and electron spins. Crucially, we find that both the nuclear spin and electron spin retain their coherence while moving the electron between quantum dots, paving the way to long range nuclear-nuclear entanglement via electron shuttlingSkinner2003Hydrogenic (). Our results establish nuclear spins in quantum dots as a powerful new resource for quantum processing.

Electrons bound to single dopant atoms or localised crystal defects strongly interact with the host donor nuclear spinsPla2013High-fidelity (); Neumann2010Single-Shot (), as well as nearby lattice nuclear spinsPla2014Coherent (); Childress2006Coherent (), due to their highly confined wavefunctions. In quantum dots, on the other hand, the electron wavefunction typically overlaps with many nuclear spins, leading to undesired effects such as loss of coherence and spin relaxationJohnson2005Tripletsinglet (); Chekhovich2013Nuclear (). In silicon metal–oxide–semiconductor quantum dots, however, the strong confinement of the electrons against the interface, together with the possibility of small gate dimensions, result in a relatively small electron wavefunctionYang2013Spin-valley (); Ruskov2018Electron (), see Fig. 1d. This leads to strong hyperfine interactions with only a few nuclei when using isotopically-enriched base material. Indeed, simulating the distribution of expected hyperfine couplingsAssali2011Hyperfine () in such a quantum dot with a 10 nm diameter and 800 ppm nuclei, we may expect two to three nuclei per quantum dot that have a hyperfine coupling larger than 100 kHz, and a maximally possible hyperfine coupling of 700 kHz.

Figure 1: Resolvable hyperfine coupling in a silicon metal-oxide quantum dot — a, Scanning electron micrograph of a device nominally identical to the one measured. The device consists of a double quantum dot accumulated under G1,G2, laterally confined by confinement gate C. GT controls the tunnel coupling to a nearby electron reservoir (RES). Electron occupation can be determined using a nearby single electron transistor (SET). Electron or nuclear spin resonance can be driven via the on-chip antenna (MW). Details of the device layout are described in Huang et al.Huang2018Fidelity (). b, (left) When monitoring the electron spin resonance (ESR) centre frequency, extracted by fitting a Gaussian to repeated ESR frequency scans, binary jumps can be observed on an hour-timescale. (right) A histogram of the centre frequencies reveals the presence of a coupled nucleus (green spin), with hyperfine coupling  kHz, and a second  kHz coupled nucleus (red spin). Histogram bin-width is 40 Hz. c, Schematic representation of the device, highlighting the formation of quantum dots QD1,QD2 at the interface under gates G1,G2. d, A close-up schematic of the interface region, showing the small (approximately 10 nm diameter), electron wavefunction with vertical valley oscillations (red, blue lobes), overlapping with a nucleus (green spin). The expected number of nuclei for the particular crystal plane shown is approximately 1.5.

In this work we experimentally investigate the effect of individual nuclear spins on the operation of a double quantum dot device (Fig. 1), that was previously characterised in Ref. Huang2018Fidelity (). The quantum dots QD1, QD2 can be completely emptied, and single electrons can be loaded from the nearby electron reservoir. Using an external magnetic field  T to split the electron spin eigenstates by 39 GHz allows spin readout via the spin-selective unloadingElzerman2004Single-shot () of an electron from QD2. Furthermore, a single electron can be transferred between QD1 and QD2, while maintaining its spin polarisationHuang2018Fidelity (); Baart2016Single-spin (). Electron spin resonance (ESR) pulses applied to an on-chip microwave antenna allow coherent manipulations of the electron spin, with intrinsic spin transition linewidths around 50 kHzVeldhorst2014addressable (). When monitoring the spin resonance frequency () for an electron loaded in QD1 (with QD2 empty) over extended periods of time, discrete jumps can be observed (Fig. 1b, left). Indeed, the histogram in Fig. 1b (right) suggests the presence of two distinct two-level systems, resulting in shifts in of approximately 120 kHz and 500 kHz.

Figure 2: nuclear spin qubit control and readout — a, Double quantum dot charge stability diagram as a function of voltages , applied to gates G1,G2. Shown are the electron occupation numbers (N1,N2), and voltage operation points used throughout for operating with a single electron in dot QD1 or QD2 as well as for the electron spin read-out (read). b, Energy levels of the joint electron-nuclear spin system, for a single nuclear spin coupled to quantum dot QD1. When QD1 is empty (right), the transition frequencies correspond to the bare Larmor frequencies and . When the electron is loaded onto QD1 (left), the hyperfine interaction causes each frequency to split , , depending on the state of the other spin. c, In order to detect the state of the nuclear spin, we compare the electron inversion probability around and . We can apply NMR pulses with QD1 unloaded, as in diagram c, or, as shown in diagram d, with QD1 loaded with a spin- electron. By applying an adiabatic ESR inversion (pulses shown in dotted square) we can also load QD1 with a spin- electron. e, In the unloaded case, schematically shown on (top), mapping the probability that the electron spin resonance frequency has switched between and as a function of applied NMR frequency and duration, we find, (bottom), coherent oscillations of the nuclear spin. f, Loading a spin- electron, schematically shown on (top right), we find, (right), the nuclear resonance frequency has shifted by . If we first flip the electron spin to , (top left), we find, (left), the nuclear frequency at , as expected. g-j, We perform nuclear Ramsey (g,h) and Hahn echo (i,j) sequences with the electron unloaded (g,i) and loaded on QD1 (h,j), in order to characterise the nuclear coherence properties. We find  ms,  ms,  ms and  ms. Ramsey sequences are performed detuned in order to accurately determine the decay. Above values correspond to a 1 hour integration time, see Extended Data Table I for details.

In order to determine whether the two-level fluctuations can be attributed to nuclear spins, we focus our attention on the 500 kHz shift. We first apply a radiofrequency (RF) tone with quantum dot QD1 unloaded, and then check whether the ESR frequency has shifted, by repeatedly probing the inversion probability around  kHz or  kHz, with the average ESR frequency. In Fig. 2e we show the probability that the electron spin resonance frequency has switched between and after applying an RF pulse with varying frequency and duration, and find coherent oscillations centred around  MHz. This frequency corresponds to a gyromagnetic ratio of  MHz/T (where the uncertainty comes from the accuracy with which we can determine the applied field ), consistent with the bulk gyromagnetic ratioPla2014Coherent () of  MHz/T. We therefore conclude that the electron in quantum dot QD1 couples to a nucleus with hyperfine coupling  kHz. We describe the joint spin system by a Hamiltonian of the form

(1)

where , are the electron and spin operators and  GHz/T, the electron gyromagnetic ratio. Here, the interaction is dominated by the contact hyperfine term, with dipole-dipole terms expected to be several orders of magnitude smallerAssali2011Hyperfine (). The Hamiltonian results in energy eigenstates shown in Fig. 2b, with both electron and nuclear spin transitions splitting by A when the electron is loaded onto QD1. Indeed, repeating the same experiment as above, but applying the nuclear magnetic resonance (NMR) pulses after loading QD1 with a spin-down electron, we find the NMR frequency has shifted to , see Fig. 2f (right peak), yielding an accurate measurement of  kHz at this control point. Finally, we repeat the experiment once more, where we load QD1 with an electron with spin up (by applying adiabatic ESR inversion, see Methods), and confirm , see Fig. 2f (left peak). In Extended Data Fig. 1 we present results for another nuclear spin coupled to the electron spin in quantum dot QD2.

Having confirmed the ability to controllably address individual nuclear spins, we proceed to characterise a qubit encoded in this new resource. As observed from the interval between jumps in Fig. 1b, the nuclear spin lifetime (while repeatedly probing the electron resonance frequency, a process which likely determines that flip rateZhao2018Coherent ()) extends to tens of minutes. By fitting an exponential decay to the intervals between the 500 kHz and 120 kHz jumps for the data in Fig. 1b, we find  hours and  minutes. This means we can perform multiple quantum-non-demolition measurements of the nuclear spin state to boost the nuclear spin state readout fidelityPla2013High-fidelity (); Neumann2010Single-Shot (). A simple simulation (Methods) for repeated readouts, taking into account the 8 ms measurement cycle and the electron spin readout visibility of 76% results in an optimal number of readouts , and an obtainable nuclear spin readout infidelity of . In this work we limit to 20, resulting in a measured nuclear spin readout fidelity of 99.8% for the dataset in Fig. 3, see Methods for details. We determine the nuclear spin coherence times by performing nuclear Ramsey and Hahn-echo sequences, with the electron unloaded (Fig. 2g,i) and loaded (see Fig. 2h,j). We find nuclear coherence times between two and three orders of magnitude longer than those measured for the electron in this deviceHuang2018Fidelity (), but shorter than previously measured for nuclear spins coupled to donors in enriched siliconMuhonen2014Storing (), possibly due to the closer proximity of the device surface. A full overview of the measured coherence times is presented in Extended Data Table I.

Figure 3: Demonstration of entanglement between nuclear and electron spin state — a, Pulse sequence used to implement the quantum circuit in b, to prepare the maximally entangled Bell state , followed by projection onto the X-Y plane of both qubits and readout of the joint electron-nuclear spin state. c,d, Varying the nuclear and electron projection phases , respectively, we observe oscillations of the two-qubit parity, as expected for the initialised Bell state . There is a clear difference in observed amplitude and phase for the electron spin- readout probabilities. The phase offset is caused by an AC stark shift induced by the off-resonant conditional ESR pulses used in the final state projection, while the reduced amplitude is a consequence of the asymmetric electron spin readout fidelity for electron spin- () versus spin- (). Solid lines show the result of two sinusoidal fits, one for electron spin- (red, black) and one for electron spin- (green,blue). The data in c,d, and Extended Data Fig. 3a,b, are jointly fit, by including phase offsets depending on the initial and final nuclear spin states. e, We characterise the Bell state initialisation fidelity, by measuring the joint state in bases XX, YY, ZZ. For the ZZ basis, the two-qubit state is measured without applying the projection pulses in b, while for the XX, YY bases we use the phases calibrated from c,. Shown are the joint readout probabilities, corrected for final electron readout fidelity only. See Main text for details.

In Fig. 3, we use coherent control to prepare entangled states of the joint electron-nuclear two-qubit system. We perform all operations with the electron loaded and construct the required unconditional rotations from two consecutive conditional rotationsDehollain2016Bells (), see Fig. 3a,b. After careful calibration of the AC Stark shifts induced by the off-resonant conditional ESR pulses (Fig. 3c,d), we characterise the prepared Bell state fidelity by measuring the two-qubit expectation values , and of the joint x,y,z-Pauli operator on the nucleus and electron, respectively. We correct the two-qubit readout probabilities for final electron readout errors only, which we calibrate by interleaving the entanglement measurement with readout fidelity characterisations (Methods). Note that because the nuclear spin is initialised by measurement here, we obtain an identical dataset corresponding to initial nuclear state , which we show in Extended Data Fig. 3. We find an average Bell state preparation fidelity of .

Our entanglement protocol is affected by several errors. By simulating our protocol, we estimate that the dominant source of error is the electron (), followed by the uncontrolled 120 kHz coupled nucleus, observed in Fig. 1b (5%). Depending on the state of this nuclear spin, detuned ESR pulses cause an unknown phase shift. Other contributions include pulse duration calibration errors () and the reduced NMR control fidelity with QD1 loaded (3%).

A unique feature of our quantum-dot-coupled nuclear spin qubit is the large ratio of interdot tunnel coupling to hyperfine coupling , so that , with  GHz in this deviceHuang2018Fidelity (). We should therefore be able to cleanly and adiabatically load and unload the electron into a neighbouring dot, without disturbing the nuclear spin state. Although this has been envisioned for donors coupled to quantum dots in siliconKane1998silicon-based (); Wolfowicz201629 (); Harvey-Collard2017Coherent (), the more tightly confining donor potential may result in insufficient control over the loading process. We can experimentally verify that the adiabatic transfer of an electron preserves the nuclear spin coherence with the pulse sequence shown in Fig. 4a. The sequence comprises a nuclear Ramsey-based experiment, where the electron is loaded onto QD1 from QD2 for a varying amount of time during free precession, while keeping the total evolution time constant. As expected, the loading of the electron causes a phase accumulation set by the hyperfine strength, see Fig. 4b, but preserves the nuclear spin coherence. To quantify the phase error induced by a load-unload cycle, we perform repeated electron shuttling, see Fig. 4c-d.

Figure 4: Nuclear and electron spin coherence during electron transfer — a, We verify that the nuclear spin coherence is maintained when moving the electron from quantum dot QD2 to QD1, by performing a non-detuned nuclear Ramsey experiment where we load the electron onto QD1 for a time during free precession, while keeping the total precession time constant. b, With QD1 loaded the nuclear phase evolution is altered relative to the bare rotating frame by the hyperfine interaction, directly observable by the phase evolution as a function of (keeping  ms). The oscillation frequency yields another measurement for . The oscillation visibility is limited by the electron spin initialisation fidelity (for electron spin- the oscillations have opposite phase). c, By repeatedly loading and unloading the electron we can estimate the loss of coherence due to the loading process. d, (bottom) To quantify the retained nuclear spin coherence independent of deterministic phase shifts we measure the probability of the nuclear spin state being in the states X, −X, Y, or –Y, corresponding to a spin up result after the final Ramsey pulse phases of , , , . (top) Nuclear coherence . Treating the loading/unloading process as a dephasing channel (Methods), we find an error probability per load/unload cycle of . Here,  ms is fixed. e, In an analogous measurement, we verify the electron spin coherence is maintained while shuttling it from QD1 to QD2, by performing an electron Ramsey where the first pulse is driven with the electron in QD1, and the second pulse with the electron in QD2. Note that because of the -factor difference between quantum dots QD1 and QD2, the second pulse has a different frequency . The nuclear spin state remains fixed in this experiment. f, Electron spin- readout probability as a function of final Ramsey phase and shuttling ramp time , showing a coherent spin transfer, with  30% visibility (no correction for readout).

The nuclear phase preservation raises the prospect to entangle nuclei in separate quantum dots, mediated by the electron shuttlingSkinner2003Hydrogenic (); Fujita2017Coherent (). For that to work, the electron spin state itself must also remain coherent during transferFujita2017Coherent (). By performing an electron Ramsey experiment, where the first pulse is driven with the electron in QD1 and the second pulse is driven with the electron in QD2, see Fig. 4e,f, we demonstrate that this is indeed the case, presently with modest transfer fidelities. A detailed investigation of the coherent electron transfer and the potential to generate nuclear entanglement using electron shuttling will be the topic of future work.

The readout and control of nuclear spins coupled to quantum dots shown here presents us with a variety of future research possibilities. Firstly, the nuclear spin qubits could form the basis for a large-scale quantum processor, where initialisation, readout and multi-qubit interactions are mediated by electron spinsSkinner2003Hydrogenic (). Secondly, the nuclear spin qubits could be used as a quantum memoryFreer2017single-atom () in an electron-spin-based quantum processor. Implementations of quantum error correcting codes may benefit from integrated, long term quantum state storage. In particular, lossy or slow long-range interactions between quantum dots, for example mediated by photonic qubitsMi2018coherent (); Samkharadze2018Strong (), could be admissible if supplemented by local nuclear spin resources for memory or purificationWolfowicz201629 (); Bennett1996Purification (); Nickerson2013Topological (). Finally, the nuclear spin qubits can be used as a characterisation tool for electron spin-based qubits. For example, in the present experiment, the confirmed existence of a nucleus with 500 kHz hyperfine coupling bounds the electron wavefunction diameter to under 12 nm, a conclusion difficult to draw with purely electrostatic calculations or electronic measurements. Further characterisations of electron spin dynamics may be envisioned by mapping the electron spin state to the nuclear spin, and employing the nucleus as a high-fidelity readout toolDehollain2016Bells (). Limitations include the extended control times for the nuclear spin, as well as the effect of sample heating from long NMR pulses. Both limitations could be addressed by redesigning the RF delivery, for example using a global RF cavity. Presently, the are randomly distributed, resulting in a range of hyperfine couplings for each quantum dot. Although the probability of obtaining at least one addressable per quantum dot is very large, implantation of nuclei in (possibly further enrichedMazzocchi201999992 ()) silicon host material would allow designing an optimal interaction. This could be done via ion implantationDonkelaar2015Single (), requiring a relatively modest precision on the order of the size of the quantum dots, compared to the precision needed for direct donor-donor coupling. In summary, we have demonstrated coherent control, entanglement and high-fidelity readout of a single nuclear spin qubit, embedded in a lithographically-defined silicon quantum dot. We find that inter-dot electron tunnelling preserves the electron and nuclear spin coherence. The combination of controllable nuclear spin qubits with the long-range interactions afforded by electrons in silicon quantum dots provides a powerful new resource for quantum processing.

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.1 Acknowledgements

We thank T.D. Ladd and C. Escott for valuable discussions and feedback on the manuscript, and J.J. Pla for technical assistance. We acknowledge support from the US Army Research Office (W911NF-17-1-0198), the Australian Research Council (CE170100012), and the NSW Node of the Australian National Fabrication Facility. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorised to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. B.H. acknowledges support from the Netherlands Organisation for Scientific Research (NWO) through a Rubicon Grant. K.M.I. acknowledges support from a Grant-in-Aid for Scientific Research by MEXT.

.2 Author contributions

B.H. and W.H. performed the experiments. K.W.C. and F.E.H. fabricated the devices. K.M.I. prepared and supplied the wafer. B.H., W.H., C.H.Y., J.Y., T.T. and A.L. designed the experiments. B.H., W.H. and J.Y. analysed the data. B.H. wrote the manuscript with input from all co-authors. A.M., A.L., and A.S.D. supervised the project.

.3 Methods

.3.1 Experimental methods

Details of sample fabrication and experimental setup can be found in Huang et al.Huang2018Fidelity (). NMR pulses were generated by a secondary Agilent E8267D microwave vector signal generator, combined with the ESR pulses using a resistive combiner. Conditional ESR pulses for nuclear readout are performed via adiabatic inversions with a frequency span of  kHz to  kHz and  kHz to  kHz for reading nuclear spin , respectively. The adiabatic pulse has a duration of and a power corresponding to a 100 kHz Rabi frequency. Electron spin transfers between QD1 and QD2 are performed with a linear ramp, except where indicated otherwise.

.3.2 Nuclear spin readout fidelity

We model the repetitive nuclear readout as a stochastic process, where, as a function of the number of shots , the fidelity is limited by the nuclear decay on the one hand,

(2)

with  ms the measurement time per shot,  hour, and the factor 2 comes from the fact that for each shot we read out the electron twice; once for an inversion around and once for . On the other hand the fidelity is limited by the cumulative binomial distribution representing the majority voting of M single shots:

(3)

A first order estimate for the nuclear readout is then obtained by

(4)

resulting in a minimum infidelity for , for , where we have taken the average electron spin readout fidelity recorded for the dataset in Fig. 3e.

.3.3 Nuclear-electron entanglement experiment: experimental details

For the datasets presented in Fig. 3e and Extended Data Fig. 3c, we interleave the following measurement sequences:

  1. Bell state preparation + ZZ projection

  2. Bell state preparation + XX projection

  3. Bell state preparation + YY projection

  4. Electron spin- readout characterisation for ZZ projection

  5. Electron spin- readout characterisation for ZZ projection

  6. Electron spin- readout characterisation for XX and YY projections

  7. Electron spin- readout characterisation for XX and YY projection

Prior to running each sequence, we initialise the electron spin state to , using a spin relaxation hotspot at the (0,1)-(1,0) charge transition (details in et al.Huang2018Fidelity ().). After running a sequence once, we read the state of both electron and nuclear spin (corresponding to a total of electron spin readouts). We then perform an ESR frequency check and, if necessary, calibration. The frequency check proceeds by applying a weak, resonant ESR pulse (60 kHz Rabi frequency), and fails if the spin inversion probability drops below 0.3. If the check fails for both and , the ESR frequency is recalibrated using a series of Ramsey sequences to estimate the detuning. Details of this ESR frequency calibration are described in the Supplementary Information of Huang et al.Huang2018Fidelity (). After recording 10 datapoints of sequence 1 in this manner, we switch to sequence 2, and so forth. After sequence 7 we loop back to sequence 1, until the end of the measurement. The nuclear spin initialisation is given by the readout result in the previous sequence. Total measurement time for the presented dataset was 9.5 hours, resulting in 4320 Bell state preparations.

The aim of sequence 4-7 is to record the actual average electron spin readout fidelity while recording the dataset. To estimate the spin-down readout fidelity (sequence 4,6), we apply no ESR pulses and measure the spin-down readout probability. To estimate the spin-up readout fidelity we apply an adiabatic inversion of the electron spin, consisting of a long 2.8 MHz wide frequency sweep centred around , with a power corresponding to a 100 kHz Rabi frequency, and measure the spin-up readout probability. Sequence 4,5 each have the same NMR pulses applied as sequence 1, but applied far detuned, in order to mimic the effect of the ZZ-projection NMR pulses on the electron spin readout fidelity, while unchanging the nuclear spin state itself. Similarly, sequence 6,7 have the same NMR pulses as sequence 2, but far detuned, to mimic the effect of XX, YY-projection pulses.

Finally, data from sequence 4-7 is also used to obtain an estimate for the nuclear spin readout fidelity: since all NMR pulses are applied off-resonant, the nuclear spin should remain unchanged. If the nuclear spin is read out differently after running sequence 4-7, this indicates a readout error has occurred. We find 5 readout errors in 3200 nuclear spin readouts, identified as such by a single outcome being different in a sequence of 10. The readout fidelity is estimated as the fraction of readout errors.

Bar plots shown in Fig. 3e and Extended Data Fig. 3c are corrected using their respective electron spin readout fidelity characterisation, using direct inversion. We estimate the Bell state fidelity using , where , , for nuclear spin- initialised data (Fig. 3e), and , for nuclear spin- initialised data (Extended Data Fig. 3c).

.3.4 Nuclear-electron entanglement: error analysis

Using the two-spin Hamiltonian, eq. (1), with two control fields and , and taking the secular and rotating wave approximation, we perform a time evolution simulation to estimate the effects of various noise sources on the nuclear-electron Bell state fidelity. We simulate the exact control sequences used in the experiment. The simulation calculates the operator at any specific time resulting in a final operator . We incorporate quasi-static noise along , and directions following a Gaussian distribution with standard deviation of , , respectively, and repeat the simulation for 1000 times to obtain the average final measurement probabilities. We use values  ms,  ms, . The value for has a large uncertainty, ranging from 8 to depending on the exact ESR frequency feedback settings and intervalHuang2018Fidelity (). To simulate the effect of the uncontrolled 120 kHz coupled spin, we estimate the probability that the nuclear spin flips within the time between ESR frequency checks, resulting in an unnoticed frequency shift. Using  min, and an average time between ESR frequency checks of 40 seconds, we find a probability of 7% of running the entanglement sequence with 120 kHz detuned ESR pulses. Finally, to simulate the effect of pulse calibration errors, we estimate our pulse-length calibration is accurate within 5%. Error percentages quoted in the Main text are the reduction in final Bell state fidelity resulting from incorporating the corresponding error mechanism only, with all other error mechanisms turned off in the simulation.

.3.5 Coherent loading dephasing analysis

We model the effect of transferring the electron between QD1 and QD2 on the nuclear spin state as a dephasing channel

(5)

This model yields an exponentially decaying off-diagonal matrix element magnitude as a function of channel transfers , , which is the measured coherence defined in the caption of Fig. 4d,e.

.4 Extended Data

Extended Data Figure 1: Ramsey and Hahn echo measurement in the (0,0) charge state — a, Ramsey measurement and b, Hahn echo measurement with nuclear pulses and free evolution time in charge configuration (0,0). Resulting values  ms and  ms are within error to those obtained for the unloaded-(0,1) charge configuration, see Fig. 2g,i.
Extended Data Figure 2: Second qubit coupled to quantum dot QD2 — a, NMR frequency scan with QD2 loaded, charge configuration (0,1), with a spin down electron reveals a nuclear spin coupled by  kHz. Note that the nuclear spin readout contrast is reduced due to the small hyperfine splitting. b, Loaded Ramsey measurement yields  ms, for 1 hour integration time. c, Loaded Hahn echo measurement yields  ms. d, unloaded, charge state (0,0), Hahn echo yields  ms.
Extended Data Figure 3: Nuclear-electron entanglement data for opposite nuclear spin initialisation — a,b, As expected, for a nuclear spin--initialised state, varying the nuclear and electron projection phases , respectively, we observe oscillations with opposite phase compared to those for a nuclear spin--initialised state, compare Fig. 3c,d. c, Accordingly, XX and YY projections have opposite parity, compare Fig. 3e.
Nuclear spin
in quantum dot
Charge state
Hyperfine
magnitude
Ramsey Hahn echo
Exponent Int. time Figure Figure
QD1 (0,1) - unloaded 450 kHz  ms 3.2 hrs Fig. 2g  ms Fig. 2i
QD1 (1,0) - loaded 450 kHz  ms 1.1 hrs Fig. 2h  ms Fig. 2j
QD1 (0,0) - unloaded 450 kHz  ms 7 hrs Ext. 1a  ms Ext. 1b
QD2 (0,1) - loaded 90 kHz  ms 6.3 hrs Ext. 2b  ms Ext. 2c
QD2 (0,0) - unloaded 90 kHz Not measured  ms Ext. 2d
Extended Data Table I: Overview of nuclear coherence times — Details and fitted values for all Ramsey and Hahn echo sequences performed on two nuclear spins, one in quantum dot QD1, one in QD2, for different charge states. Ramsey values are fits to a sinusoidal function with envelope decay . Int. time indicates total integration time for the measurement. Hahn echo values are fits to an exponential decay . Figure panels displaying each measurement are indicated. Note that for the -values given in the captions of Fig. 2 and Extended Data Figures 1,2, the integration times are all limited to 1 hour, for comparison.
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