A silicon quantum-dot-coupled nuclear spin qubit

TL;DR

This paper demonstrates the control, readout, and entanglement of silicon-29 nuclear spins in quantum dots, combining long coherence times with scalable quantum dot technology, and enabling long-range nuclear spin interactions.

Contribution

It shows that hyperfine interactions in silicon quantum dots are strong enough for nuclear spin control, enabling scalable quantum computing with long coherence times.

Findings

High-fidelity nuclear spin readout and control achieved.

Entanglement demonstrated between nuclear and electron spins.

Electron shuttling preserves spin coherence, enabling long-range interactions.

Abstract

Single nuclear spins in the solid state have long been envisaged as a platform for quantum computing, due to their long coherence times and excellent controllability. Measurements can be performed via localised electrons, for example those in single atom dopants or crystal defects. However, establishing long-range interactions between multiple dopants or defects is challenging. Conversely, in lithographically-defined quantum dots, tuneable interdot electron tunnelling allows direct coupling of electron spin-based qubits in neighbouring dots. Moreover, compatibility with semiconductor fabrication techniques 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 shuttling. Our results establish nuclear spins in quantum dots as a powerful new resource for quantum processing.

Figures

b, the nuclear spin lifetime (while repeatedly probing the electron resonance frequency, a process which likely affects that flip rate) extends to tens of minutes. By fitting an exponential decay to the intervals between the 450 kHz and 120 kHz jumps for the data in Fig. 1b, we find $T_{1}^{\mathrm{(450kHz)}}=1.0\pm 0.5$ hours and $T_{1}^{\mathrm{(120kHz)}}=10.0\pm 0.6$ minutes. (see Supplementary Fig. S1 for details). This means we can perform multiple quantum-non-demolition measurements of the nuclear spin state to boost the nuclear spin state readout fidelity[Pla2013High-fidelity ](#bib.bib

b, with both electron and nuclear spin transitions splitting by A when the electron is loaded onto QD1. 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 $f_{\mathrm{n}}^{\downarrow}=f_{\mathrm{n}}^{0}+|A|/2$, see Fig. 2f (right peak), yielding an accurate measurement of $A=-448.5\pm 0.1$ 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 c

e 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 $F=F_{\langle ZZ\rangle}/2+F_{\langle YY\rangle}/2+F_{\langle XX\rangle}/2-1/2$, where $F_{\langle ZZ\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$, $F_{\langle YY\rangle}=p_{\Downarrow\uparrow}+p_{\Uparrow\downarrow}$, $F_{\langle XX\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$ for nuclear spin-$\Downarrow$ initialised data (Fig. 3e), and $F_{\langle YY\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow

a. 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.

b, the nuclear spin lifetime (while repeatedly probing the electron resonance frequency, a process which likely affects that flip rate) extends to tens of minutes. By fitting an exponential decay to the intervals between the 450 kHz and 120 kHz jumps for the data in Fig. 1b, we find $T_{1}^{\mathrm{(450kHz)}}=1.0\pm 0.5$ hours and $T_{1}^{\mathrm{(120kHz)}}=10.0\pm 0.6$ minutes. (see Supplementary Fig. S1 for details). This means we can perform multiple quantum-non-demolition measurements of the nuclear spin state to boost the nuclear spin state readout fidelity[Pla2013High-fidelity ](#bib.bib

b, with both electron and nuclear spin transitions splitting by A when the electron is loaded onto QD1. 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 $f_{\mathrm{n}}^{\downarrow}=f_{\mathrm{n}}^{0}+|A|/2$, see Fig. 2f (right peak), yielding an accurate measurement of $A=-448.5\pm 0.1$ 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 c

e 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 $F=F_{\langle ZZ\rangle}/2+F_{\langle YY\rangle}/2+F_{\langle XX\rangle}/2-1/2$, where $F_{\langle ZZ\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$, $F_{\langle YY\rangle}=p_{\Downarrow\uparrow}+p_{\Uparrow\downarrow}$, $F_{\langle XX\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$ for nuclear spin-$\Downarrow$ initialised data (Fig. 3e), and $F_{\langle YY\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow

a. 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.

b, the nuclear spin lifetime (while repeatedly probing the electron resonance frequency, a process which likely affects that flip rate) extends to tens of minutes. By fitting an exponential decay to the intervals between the 450 kHz and 120 kHz jumps for the data in Fig. 1b, we find $T_{1}^{\mathrm{(450kHz)}}=1.0\pm 0.5$ hours and $T_{1}^{\mathrm{(120kHz)}}=10.0\pm 0.6$ minutes. (see Supplementary Fig. S1 for details). This means we can perform multiple quantum-non-demolition measurements of the nuclear spin state to boost the nuclear spin state readout fidelity[Pla2013High-fidelity ](#bib.bib

b, with both electron and nuclear spin transitions splitting by A when the electron is loaded onto QD1. 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 $f_{\mathrm{n}}^{\downarrow}=f_{\mathrm{n}}^{0}+|A|/2$, see Fig. 2f (right peak), yielding an accurate measurement of $A=-448.5\pm 0.1$ 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 c

e 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 $F=F_{\langle ZZ\rangle}/2+F_{\langle YY\rangle}/2+F_{\langle XX\rangle}/2-1/2$, where $F_{\langle ZZ\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$, $F_{\langle YY\rangle}=p_{\Downarrow\uparrow}+p_{\Uparrow\downarrow}$, $F_{\langle XX\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$ for nuclear spin-$\Downarrow$ initialised data (Fig. 3e), and $F_{\langle YY\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow

Tables

Table 1. Extended Data Table I: Overview of nuclear coherence times — Details and fitted values for all Ramsey and Hahn echo sequences performed on two Si 29 superscript Si 29 {}^{29}\mathrm{Si} 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 exp ⁡ [ − ( τ / T 2 ∗ ) α ] superscript 𝜏 superscript subscript 𝑇 2 𝛼 ~{}\exp\left[-(\tau/T_{2}^{*})^{\alpha}\right] . Int. time indicates total integration time for the measurement. Hahn echo values are fits to an exponential decay exp ⁡ [ − 2 ​ τ / T 2 Hahn ] 2 𝜏 superscript subscript 𝑇 2 Hahn ~{}\exp\left[-2\tau/T_{2}^{\mathrm{Hahn}}\right] . Figure panels displaying each measurement are indicated. Note that for the T 2 ∗ superscript subscript 𝑇 2 T_{2}^{*} -values given in the captions of Fig. 2 and Extended Data Figures 2,3, the integration times are all limited to 1 hour, for comparison.

Nuclear spin in quantum dot	Charge state	Hyperfine magnitude	Ramsey				Hahn echo
			$T_2^{\ast }$	Exponent $\alpha$	Int. time	Figure	$T_2^{\mathrm{Hahn}}$	Figure
QD1	(0,1) - unloaded	450 kHz	$6.6\pm 0.2$ ms	$2.11\pm 0.21$	3.2 hrs	Fig. 2g	$16.3\pm 2.4$ ms	Fig. 2i
QD1	(1,0) - loaded	450 kHz	$2.9\pm 0.7$ ms	$0.91\pm 0.25$	1.1 hrs	Fig. 2h	$22.8\pm 4.1$ ms	Fig. 2j
QD1	(0,0) - unloaded	450 kHz	$5.9\pm 0.2$ ms	$1.54\pm 0.39$	7 hrs	Ext. 2a	$13.1\pm 1.5$ ms	Ext. 2b
QD2	(0,1) - loaded	180 kHz	$21.3\pm 2.0$ ms	$1.74\pm 0.16$	6.3 hrs	Ext. 3b	$42.2\pm 10.6$ ms	Ext. 3c
QD2	(0,0) - unloaded	180 kHz	Not measured				$40.5\pm 13.0$ ms	Ext. 3d

Equations

$$H=-B_{\mathrm{ext}}({\gamma }_e{S}_z+{\gamma }_{\mathrm{Si}}{I}_z)+A(S\cdot I),$$

$$F_{T_1}=\exp \left[-2M{t}_{\mathrm{shot}}/T_1\right],$$

$$F_{\mathrm{shot}}=\sum _{k=0}^{2M/2}\left(\genfrac{}{}{0ex}{}{2M}{k}\right)(1-F_{\mathrm{e}}^{\mathrm{avg}}{)}^k(F_{\mathrm{e}}^{\mathrm{avg}}{)}^{2M-k}.$$

$$F_{\mathrm{n}}\approx (1-F_{T_1})F_{\mathrm{shot}}+F_{T_1}(1-F_{\mathrm{shot}}),$$

$$\rho =\left(\begin{array}{cc}{\rho }_{00}& {\rho }_{01}\\ {\rho }_{10}& {\rho }_{11}\end{array}\right)\to {\rho }^{\prime }=(1-p_{\mathrm{err}})\rho +p_{\mathrm{err}}\left(\begin{array}{cc}{\rho }_{00}& 0\\ 0& {\rho }_{11}\end{array}\right).$$

Full text

††thanks: These authors contributed equally††thanks: These authors contributed equally

A silicon quantum-dot-coupled nuclear spin qubit

Bas Hensen

  

Wister Wei 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

  

Thaddeus D. Ladd

HRL Laboratories, LLC, 3011 Malibu Canyon Rd., Malibu, CA, 90265, USA

  

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

Single 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 dopantsPla2013High-fidelity ; Pla2014Coherent or crystal defectsChildress2006Coherent ; Neumann2010Single-Shot ; Abobeih2018One-second . However, establishing long-range interactions between multiple dopants or defects is challengingBernien2013Heralded ; Tosi2017Silicon . 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 ; Huang2019Fidelity . 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 or defect nuclear spinsPla2013High-fidelity ; Neumann2010Single-Shot , as well as nearby lattice nuclear spinsPla2014Coherent ; Childress2006Coherent , due to their highly confined wavefunctions. In quantum dots, the electron wavefunction is less confined and typically overlaps with many more 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 $\mathrm{Si-SiO_{2}}$ interface, together with the possibility of small gate dimensions, result in a relatively small electron wavefunctionYang2013Spin-valley , see Fig. 1a,b. This leads to strong hyperfine interactions, and when using isotopically enriched ${}^{28}\mathrm{Si}$ base material, the number of interacting ${}^{29}\mathrm{Si}$ nuclei may be only a few. Simulations of the distribution of expected hyperfine couplingsAssali2011Hyperfine in a quantum dot with a 8 nm wavefunction diameter and 800 ppm ${}^{29}\mathrm{Si}$ nuclei indicate an expectation of two to three ${}^{29}\mathrm{Si}$ nuclei per quantum dot that have a resolvable hyperfine coupling ($\geq 100$ kHz), and a maximally possible hyperfine coupling of approximately 400 kHz; see Extended Data Fig. 1.

In this work we experimentally investigate the effect of individual nuclear spins on the operation of a double quantum dot device (Fig. 1c), that was previously characterised in Ref.Huang2019Fidelity . The quantum dots QD1 and QD2 can be completely emptied, and single electrons can be loaded from the nearby electron reservoir. Using an external magnetic field $B_{\mathrm{ext}}=1.42\pm 0.04$ 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 polarisationVeldhorst2015two-qubit ; Huang2019Fidelity . 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 ($f_{\mathrm{e}}$) for an electron loaded in QD1 (with QD2 empty) over extended periods of time, discrete jumps can be observed (Fig. 1d, left). Indeed, the histogram in Fig. 1d (right) suggests the presence of two distinct two-level systems, resulting in shifts in $f_{\mathrm{e}}$ of approximately 120 kHz and 450 kHz.

In order to determine whether the two-level fluctuations can be attributed to ${}^{29}\mathrm{Si}$ nuclear spins, we focus our attention on the 450 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 electron spin inversion probability around $f_{\mathrm{e}}^{\Uparrow}=f_{\mathrm{e}}^{0}-225$ kHz or $f_{\mathrm{e}}^{\Downarrow}=f_{\mathrm{e}}^{0}+225$ kHz, with $f_{\mathrm{e}}^{0}$ the average ESR frequency. In Fig. 2e we show the probability $P_{\mathrm{flip}}$ that the electron spin resonance frequency has switched between $f_{\mathrm{e}}^{\Uparrow}$ and $f_{\mathrm{e}}^{\Downarrow}$ after applying an RF pulse with varying frequency and duration, and find coherent oscillations centred around $f_{\mathrm{n}}^{0}=11.9078$ MHz. This frequency corresponds to a gyromagnetic ratio with magnitude $8.37\pm 0.15$ MHz/T (where the uncertainty comes from the accuracy with which we can determine the applied field $B_{\mathrm{ext}}$), consistent with the bulk ${}^{29}\mathrm{Si}$ gyromagnetic ratioPla2014Coherent of $\gamma_{\mathrm{Si}}=-8.458$ MHz/T. We therefore conclude that the electron in quantum dot QD1 couples to a ${}^{29}\mathrm{Si}$ nucleus with hyperfine coupling $|A|=|f_{\mathrm{e}}^{\Uparrow}-f_{\mathrm{e}}^{\Downarrow}|\approx 450$ kHz. We describe the joint spin system by a Hamiltonian of the form

[TABLE]

where $S$, $I$ are the electron and ${}^{29}\mathrm{Si}$ spin operators and $\gamma_{e}=-28$ 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. 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 $f_{\mathrm{n}}^{\downarrow}=f_{\mathrm{n}}^{0}+|A|/2$, see Fig. 2f (right peak), yielding an accurate measurement of $A=-448.5\pm 0.1$ 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 $f_{\mathrm{n}}^{\uparrow}=f_{\mathrm{n}}^{0}-|A|/2$, see Fig. 2f (left peak). In Extended Data Fig. 3 we present results for another ${}^{29}\mathrm{Si}$ nuclear spin coupled to the electron spin in quantum dot QD2 that has hyperfine $A_{\mathrm{QD2}}=-179.8\pm 0.2$ kHz. Electrostatic modelling based on the device gate geometry indicates an electron wavefunction diameter of around 8 nm, which is consistent with the hyperfine couplings observed in QD2, however the coupling observed for the ${}^{29}\mathrm{Si}$ in QD1, while possible for a 7 nm diameter, suggests that an additional disorder potential may have reduced the size of this dot wavefunction.

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 affects that flip rate) extends to tens of minutes. By fitting an exponential decay to the intervals between the 450 kHz and 120 kHz jumps for the data in Fig. 1b, we find $T_{1}^{\mathrm{(450kHz)}}=1.0\pm 0.5$ hours and $T_{1}^{\mathrm{(120kHz)}}=10.0\pm 0.6$ minutes. (see Supplementary Fig. S1 for details). 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 $M$ 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 $M_{\mathrm{opt}}=26$, and an obtainable nuclear spin readout fidelity of 99.99%. In this work we limit $M$ 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 device ($T_{2}^{*\mathrm{,e}}\approx 15\mathrm{\mu s}$Huang2019Fidelity ), 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. In Supplementary Note 1, we discuss possible sources of dephasing resulting in the measured coherence times.

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 $\langle\mathrm{XX}\rangle$, $\langle\mathrm{YY}\rangle$ and $\langle\mathrm{ZZ}\rangle$ 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 $\left|\Uparrow\right\rangle$, which we show in Extended Data Fig. 3. We find an average Bell state preparation fidelity of $73.0\pm 1.9\%$.

Our entanglement protocol is affected by several errors. By simulating our protocol, we estimate that the dominant source of error is the electron $T_{2}^{*\mathrm{,e}}$ ($\approx 10\%$), followed by the uncontrolled 120 kHz coupled ${}^{29}\mathrm{Si}$ 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 ($\approx 2\%$) 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 tc to hyperfine coupling A, so that $|t_{c}|\gg|A|\gg 1/T_{2}^{*\mathrm{,e}}$, with $|t_{c}|$ on the order of 1 GHz in this deviceHuang2019Fidelity . We should therefore be able to accurately and adiabatically control the movement of an electron charge between neighbouring dots, inducing a predicable phase shift to the nuclear spin state. Importantly, the ${}^{29}\mathrm{Si}$ atom is isoelectronic with the rest of the crystal: its presence does not introduce any additional electrostatic perturbations. This is in contrast with the case of donors, where the nuclear spin resource is accompanied by a sharp Coulomb binding potential, which must be accounted for when designing the charge shuttling protocolsWolfowicz201629 ; Pica2016Surface ; Harvey-Collard2017Coherent . 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.

The nuclear phase preservation raises the prospect to entangle nuclei in separate quantum dots, mediated by the electron shuttlingSkinner2003Hydrogenic . For that to work, the electron spin state itself must also remain coherent during transferFujita2017Coherent . By performing an electron Ramsey experiment, where the first $\pi/2$ pulse is driven with the electron in QD1 and the second $\pi/2$ pulse is driven with the electron in QD2, see Fig. 4e,f, we demonstrate that electrons do indeed preserve coherence, presently with modest transfer fidelities.

The readout and control of nuclear spins coupled to quantum dots shown here presents us with a variety of future research possibilities. First, 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 . Second, 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 microwave 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 450 kHz hyperfine coupling bounds the electron wavefunction diameter to under 8 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 effects of long NMR pulses on electron spin readout (see Supplementary Note 2). Both limitations could be addressed by redesigning the RF delivery, for example using a global RF cavity. Presently, the ${}^{29}\mathrm{Si}$ are randomly distributed, resulting in a range of hyperfine couplings for each quantum dot. Although the probability of obtaining at least one addressable ${}^{29}\mathrm{Si}$ per quantum dot is large (Extended Data Fig. 1), implantation of ${}^{29}\mathrm{Si}$ nuclei in (possibly further enrichedMazzocchi201999992 ) silicon host material would allow the design of 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, which is much more forgiving than the precision needed for direct donor-donor coupling.

In summary, we have demonstrated coherent control, entanglement and high-fidelity readout of a single ${}^{29}\mathrm{Si}$ 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.

.1 Acknowledgements

We thank C. Escott, P. Harvey-Collard, V.V. Dobrovitski 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), Silicon Quantum Computing Proprietary Limited, 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 28Si 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. T.D.L. performed hyperfine and coherence simulations. 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 alHuang2019Fidelity . NMR pulses were generated by a secondary 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 $f_{\mathrm{e}}^{\Uparrow}-300$ kHz to $f_{\mathrm{e}}^{\Uparrow}+50$ kHz and $f_{\mathrm{e}}^{\Downarrow}-50$ kHz to $f_{\mathrm{e}}^{\Downarrow}+300$ kHz for reading nuclear spin $\Uparrow$, $\Downarrow$ respectively. The adiabatic pulse has a duration of $650~{}\mathrm{\mu s}$ and a power corresponding to a 100 kHz Rabi frequency. Electron spin transfers between QD1 and QD2 are performed with a $1\mathrm{\mu s}$ 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 $M$, the fidelity is limited by the nuclear $T_{1}$ decay on the one hand,

[TABLE]

with $t_{\mathrm{shot}}=8$ ms the measurement time per shot, $T_{1}=1$ hour, and the factor 2 comes from the fact that for each shot we read out the electron twice; once for an inversion around $f_{\mathrm{e}}^{\Uparrow}$ and once for $f_{\mathrm{e}}^{\Downarrow}$. On the other hand the fidelity is limited by the cumulative binomial distribution representing the majority voting of M single shots:

[TABLE]

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

[TABLE]

resulting in a minimum infidelity $1-F_{\mathrm{n}}=10^{-4}$ for $M_{\mathrm{opt}}=26$, for $F_{\mathrm{e}}^{\mathrm{avg}}=76.5\%$, 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:

Bell state preparation + ZZ projection 2. 2.

Bell state preparation + XX projection 3. 3.

Bell state preparation + YY projection 4. 4.

Electron spin-$\downarrow$ readout characterisation for ZZ projection 5. 5.

Electron spin-$\uparrow$ readout characterisation for ZZ projection 6. 6.

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

Electron spin-$\uparrow$ readout characterisation for XX and YY projection

Prior to running each sequence, we initialise the electron spin state to $\downarrow$, using a spin relaxation hotspot at the (0,1)-(1,0) charge transition (details in et al.Huang2019Fidelity ). After running a sequence once, we read the state of both electron and nuclear spin (corresponding to a total of $2M+1$ 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 $f_{\mathrm{e}}^{\Uparrow}$ and $f_{\mathrm{e}}^{\Downarrow}$, 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 alHuang2019Fidelity . 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-$\downarrow$ readout fidelity (sequence 4,6), we apply no ESR pulses and measure the spin-$\downarrow$ readout probability. To estimate the spin-$\uparrow$ readout fidelity we apply an adiabatic inversion of the electron spin, consisting of a $650~{}\mathrm{\mu s}$ long 2.8 MHz wide frequency sweep centred around $f_{e}^{0}$, with a power corresponding to a 100 kHz Rabi frequency, and measure the spin-$\uparrow$ 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 not changing 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. For the ZZ-projection we find readout fidelities $F_{e}^{\downarrow}=88.4\%$ and $F_{e}^{\uparrow}=73.3\%$, while for the XX, YY-projection we find $F_{e}^{\downarrow}=80.7\%$ and $F_{e}^{\uparrow}=67.4\%$. See also Supplementary Note 2 for a discussion of the effect of RF pulses on electron spin readout fidelity.

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 $F=F_{\langle ZZ\rangle}/2+F_{\langle YY\rangle}/2+F_{\langle XX\rangle}/2-1/2$, where $F_{\langle ZZ\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$, $F_{\langle YY\rangle}=p_{\Downarrow\uparrow}+p_{\Uparrow\downarrow}$, $F_{\langle XX\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$ for nuclear spin-$\Downarrow$ initialised data (Fig. 3e), and $F_{\langle YY\rangle}=p_{\Downarrow\downarrow}+p_{\Uparrow\uparrow}$, $F_{\langle XX\rangle}=p_{\Downarrow\uparrow}+p_{\Uparrow\downarrow}$ for nuclear spin-$\Uparrow$ initialised data (Extended Data Fig. 3c).

.3.4 Nuclear-electron entanglement: error analysis

Using the two-spin Hamiltonian, eq. (1), with two control fields $V_{\mathrm{ESR}}=|\gamma_{\mathrm{e}}|B_{1}(t)S_{x}$ and $V_{\mathrm{NMR}}=|\gamma_{\mathrm{Si}}|B_{1}(t)I_{x}$, 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 $B_{1}(t)$ used in the experiment. The simulation calculates the operator at any specific time $U_{dt}(t)=\exp\left[-2i\pi H_{\mathrm{RWA}}(t)t\right]$ resulting in a final operator $U=\prod U_{dt}(t)$. We incorporate quasi-static noise along $I_{x}$, $I_{z}$ and $S_{z}$ directions following a Gaussian distribution with standard deviation of $\frac{1}{\sqrt{2}\pi T_{2}^{\mathrm{Rabi,n}}}$, $\frac{1}{\sqrt{2}\pi T_{2}^{\mathrm{*,n}}}$, $\frac{1}{\sqrt{2}\pi T_{2}^{\mathrm{*,e}}}$ respectively, and repeat the simulation for 1000 times to obtain the average final measurement probabilities. We use values $T_{2}^{\mathrm{Rabi,n}}=1.1$ ms, $T_{2}^{\mathrm{*,n}}=T_{2}^{\mathrm{*,loaded}}=2.9$ ms, $T_{2}^{\mathrm{*,e}}=15~{}\mathrm{\mu s}$. The value for $T_{2}^{\mathrm{*,e}}$ has a large uncertainty, ranging from 8 to $22\mathrm{\mu s}$ depending on the exact ESR frequency feedback settings and intervalHuang2019Fidelity . To simulate the effect of the uncontrolled 120 kHz coupled ${}^{29}\mathrm{Si}$ spin, we estimate the probability that the nuclear spin flips within the time between ESR frequency checks, resulting in an unnoticed frequency shift. Using $T_{1}^{\mathrm{(120kHz)}}=10$ 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

[TABLE]

This model yields an exponentially decaying off-diagonal matrix element magnitude as a function of channel transfers $k$, $|\rho_{01}(k)|=1/2\exp{-k*p_{\mathrm{err}}}=1/2C(k)$, which is the measured coherence defined in the caption of Fig. 4d,e.

.4 Extended Data