Abstract
Nuclear spins were among the first physical platforms to be considered for quantum information processing^{1,2}, because of their exceptional quantum coherence^{3} and atomicscale footprint. However, their full potential for quantum computing has not yet been realized, owing to the lack of methods with which to link nuclear qubits within a scalable device combined with multiqubit operations with sufficient fidelity to sustain faulttolerant quantum computation. Here we demonstrate universal quantum logic operations using a pair of ionimplanted ^{31}P donor nuclei in a silicon nanoelectronic device. A nuclear twoqubit controlledZ gate is obtained by imparting a geometric phase to a shared electron spin^{4}, and used to prepare entangled Bell states with fidelities up to 94.2(2.7)%. The quantum operations are precisely characterized using gate set tomography (GST)^{5}, yielding onequbit average gate fidelities up to 99.95(2)%, twoqubit average gate fidelity of 99.37(11)% and twoqubit preparation/measurement fidelities of 98.95(4)%. These three metrics indicate that nuclear spins in silicon are approaching the performance demanded in faulttolerant quantum processors^{6}. We then demonstrate entanglement between the two nuclei and the shared electron by producing a Greenberger–Horne–Zeilinger threequbit state with 92.5(1.0)% fidelity. Because electron spin qubits in semiconductors can be further coupled to other electrons^{7,8,9} or physically shuttled across different locations^{10,11}, these results establish a viable route for scalable quantum information processing using donor nuclear and electron spins.
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Data availability
The experimental data that support the findings of this study are available in Figshare, https://doi.org/10.6084/m9.figshare.c.5471706. Source data are provided with this paper.
Code availability
The GST analysis was performed using a developmental version of pyGSTi that requires expertlevel knowledge of the software to install and run. A future official release of pyGSTi will support the type of analysis performed here using a simple and well documented Python script. Until this code is available, interested readers can contact the corresponding author to get help with accessing and running the existing code. Multivalley effective mass theory calculations, some of the results of which are illustrated in Fig. 1b, were performed using a fork of the code first developed in the production of ref. ^{60} that was extended to include multielectron interactions as reported in ref. ^{59}. Requests for a license for and copy of this code will be directed to points of contact at Sandia National Laboratories and the University of New South Wales, through the corresponding author. The analysis code for Bell state tomography is in Figshare, https://doi.org/10.6084/m9.figshare.c.5471706.
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Acknowledgements
We acknowledge conversations with W. Huang, R. Rahman, S. Seritan and C. H. Yang and technical support from T. Botzem. The research was supported by the Australian Research Council (grant no. CE170100012), the US Army Research Office (contract no. W911NF1710200), and the Australian Department of Industry, Innovation and Science (grant no. AUSMURI000002). We acknowledge support from the Australian National Fabrication Facility (ANFF). This material is based upon work supported in part by the iHPC facility at the University of Technology Sydney (UTS), by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research’s Quantum Testbed Pathfinder and Early Career Research Programs, and by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers (Quantum Systems Accelerator). Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the US Department of Energy’s National Nuclear Security Administration under contract DENA0003525. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of the US Department of Energy, or the US Government.
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M.T.M., V.S. and F.E.H. fabricated the device, with the supervision of A.M. and A.S.D., on an isotopically enriched ^{28}Si wafer supplied by K.M.I. A.M.J., B.C.J. and D.N.J. designed and performed the ion implantation. M.T.M. and S.A. performed the experiments and analysed the data, with A.L. and A.M.’s supervision. B.J. and A.D.B. developed and applied computational tools to calculate the electron wavefunction and the Hamiltonian evolution. A.Y. designed the initial GST sequences, with C.F.’s supervision. K.M.R., E.N., K.C.Y., T.J.P. and R.B.K. developed and applied the GST method. A.M., R.B.K., M.T.M. and S.A. wrote the manuscript, with input from all coauthors.
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Extended data figures and tables
Extended Data Fig. 1 Device layout.
Scanning electron micrograph of a device identical to the one used in this experiment. ^{31}P donor atoms are implanted in the region marked by the orange rectangle, using a fluence of 1.4 × 10^{12} cm^{−2} which results in a most probable interdonor spacing of approximately 8 nm. Four metallic gates are fabricated around the implantation region, and used to modify the electrochemical potential of the donors. A nearby SET, formed using the SET top gate and barrier gates, enables charge sensing of a single donor atom, as well as its electron spin through spintocharge conversion (Methods). The tunnel coupling between the donors and SET is tuned by the rate gate situated between the SET and donor implant region. A nearby microwave (MW) antenna is used for ESR and NMR of the donor electron and nuclear spins, respectively.
Extended Data Fig. 2 Electrical tunability of the hyperfine interaction and the electron gyromagnetic ratio.
a, Map of the SET current as a function of SET gate and fast donor gates (pulsed jointly). The white dashed line indicates the location in gate space where the 2P donor cluster changes its charge state. The third, hyperfinecoupled electron is present on the cluster in the region to the right of the line. Electron spin readout is performed at the location indicated by the pink star. b, ESR spectrum of the electron bound to the 2P cluster, acquired while the system was tuned within the blue dashed rectangle in a. The hyperfine couplings A_{1}, A_{2} are extracted from ESR frequencies as shown, namely \({A}_{1}=({\nu }_{{\rm{e}}\Uparrow \Downarrow }+{\nu }_{{\rm{e}}\Uparrow \Uparrow })/2({\nu }_{{\rm{e}}\Downarrow \Downarrow }+{\nu }_{{\rm{e}}\Downarrow \Uparrow })/2\); \({A}_{2}={\nu }_{{\rm{e}}\Uparrow \Uparrow }{\nu }_{{\rm{e}}\Uparrow \Downarrow }\). c, d, Extracted hyperfine couplings within the marked area. The data show that A_{1} decreases and A_{2} increases upon moving the operation point towards higher gate voltages and away from the donor readout position. e, A small change is also observed in the sum of the two hyperfine interactions A_{t} = A_{1} + A_{2}. f, Electrical modulation (Stark shift) of the electron gyromagnetic ratio γ_{e}, extracted from the shift of the average of the hyperfinesplit electron resonances. The ESR frequencies can be tuned with fast donor gates at the rate of \(\Delta {\nu }_{{\rm{e}}\Uparrow \Uparrow }=0.3{{\rm{MHzV}}}^{1}\); \(\Delta {\nu }_{{\rm{e}}\Uparrow \Downarrow }=5.2{{\rm{MHzV}}}^{1}\); \(\Delta {\nu }_{{\rm{e}}\Downarrow \Uparrow }=7.6{{\rm{MHzV}}}^{1}\); \(\Delta {\nu }_{{\rm{e}}\Downarrow \Downarrow }=2.4{{\rm{MHzV}}}^{1}\).
Extended Data Fig. 3 Coherence metrics of the electron spin qubit.
The columns correspond to the nuclear configurations \(\Downarrow \Downarrow \rangle \), \(\Downarrow \Uparrow \rangle \), \(\Uparrow \Downarrow \rangle \), \(\Uparrow \Uparrow \rangle \), respectively. All measurements start with the electron spin initialized in the \(\downarrow \rangle \) state. Error bars are 1σ confidence intervals. a, Electron Rabi oscillations. The measurements were performed by applying a resonant ESR pulse of increasing duration. The different Rabi frequencies f_{Rabi} on each resonance are probably due to a frequencydependent response of the onchip antenna and the cable connected to it. b, Electron spinlattice relaxation times T_{1e}. Measurements were obtained by first adiabatically inverting the electron spin to \(\uparrow \rangle \), followed by a varying wait time τ before electron readout. The observed relaxation times are nearly three orders of magnitude shorter than typically observed in singleelectron, singledonor devices^{66}, and even shorter compared to 1e–2P clusters. This strongly suggests that the measured electron is the third one, on top of two more tightlybound electrons which form a singlet spin state^{67}. We also observe a strong dependence of T_{1e} on nuclear spin configuration. c, Electron dephasing times \({T}_{2{\rm{e}}}^{\ast }\). The measurements were conducted by performing a Ramsey experiment—that is, by applying two π/2 pulses separated by a varying wait time τ, followed by electron readout. The Ramsey fringes are fitted to a function of the form \({P}_{\uparrow }(\tau )={C}_{0}+{C}_{1}\,\cos (\Delta \omega \tau +\Delta \phi )\exp [{(\tau /{T}_{2e}^{\ast })}^{2}]\), where Δω is the frequency detuning and Δϕ is a phase offset. The observed \({T}_{2{\rm{e}}}^{\ast }\) times are comparable to previous values for electrons coupled to a single ^{31}P nucleus. d, Electron Hahnecho coherence times \({T}_{2{\rm{e}}}^{{\rm{H}}}\), obtained by adding a π refocusing pulse to the Ramsey sequence. We also varied the phase of the final π/2 pulse at a rate of one period per τ = (5 kHz)^{−1}, to introduce oscillations in the spinup fraction which help improve the fitting. The curves are fitted to the same function used to fit the Ramsey fringes, with fixed Δω = 5 kHz. The measured \({T}_{2{\rm{e}}}^{{\rm{H}}}\) times are similar to previous observations for electrons coupled to a single ^{31}P nucleus.
Extended Data Fig. 4 Nuclear spin coherence times.
Panels in column 1 (2) correspond to nucleus Q1 (Q2). Error bars are 1σ confidence intervals. a, Nuclear dephasing times \({T}_{2{\rm{n}}}^{\ast }\), obtained from a Ramsey experiment. Results are fitted with a decaying sinusoid with fixed exponent factor 2 (see Extended Data Fig. 3). b, Nuclear Hahnecho coherence times \({T}_{2{\rm{n}}}^{{\rm{H}}}\). To improve fitting, oscillations are induced by incrementing the phase of the final π/2 pulse with τ at a rate of one period per (3.5 kHz)^{−1}. Results are fitted with a decaying sinusoid with fixed exponent factor 2 (see Extended Data Fig. 3). c, Dependence of \({T}_{2{\rm{n}}}^{{\rm{H}}}\) on the amplitude of an offresonance pulse. We perform this experiment to study whether a qubit, nominally left idle (or, in quantum information terms, subjected to an identity gate) is affected by the application of an RF pulse to the other qubit, at a vastly different frequency. Here, during the idle times between NMR pulses, an RF pulse is applied at a fixed frequency 20 MHz—far off resonance from both qubits’ transitions—with varying amplitude V_{RF}. The red dashed line indicates the applied RF amplitude for NMR pulses throughout the experiment. We observe a slow decrease of \({T}_{2{\rm{n}}}^{{\rm{H}}}\) with increasing V_{RF}. This is qualitatively consistent with the observation of large stochastic errors on the idle qubit, as extracted by the GST analysis in Fig. 3.
Extended Data Fig. 5 Nuclear spin quantum jumps caused by ionization shock.
The electron and nuclear spin readout relies upon spindependent charge tunnelling between the donors and the SET island. If the electron tunnels out of the twodonor system, the hyperfine interactions A_{1}, A_{2} suddenly drop to zero. If A_{1} and A_{2} include an anisotropic component (for example, due to the nonspherical shape of the electron wavefunction which results in nonzero dipolar fields at the nuclei), the ionization is accompanied by a sudden change in the nuclear spin quantization axes (‘ionization shock’), and can result in a flip of the nuclear spin state. We measure the nuclear spin flips caused by ionization shock by forcibly loading and unloading an electron from the 2P cluster every 0.8 ms. a, For qubit 1 with A_{1} = 95 MHz, the flip rate is \({{\Gamma }}_{1}=2.8\times {10}^{6}\frac{{N}_{{\rm{flip}}}}{{N}_{{\rm{ion}}}}\). b, For qubit 2 with A_{2} = 9 MHz, the flip rate is \({{\Gamma }}_{2}=4.0\times {10}^{7}\frac{{N}_{{\rm{flip}}}}{{N}_{{\rm{ion}}}}\). This means that the nuclear spin readout via the electron ancilla is almost exactly quantum nondemolition. From this data, we also extract an average time between random nuclear spin flips of 283 seconds for qubit 1, and 2,000 seconds for qubit 2. The extremely low values of Γ—comparable to those observed in singledonor systems—are the reason why we can reliably operate the two ^{31}P nuclei as highfidelity qubits.
Extended Data Fig. 6 CNOT and zeroCNOT nuclear twoqubit gates.
We perform Rabi oscillation on the control qubit followed by the application of a, zCNOT or b, CNOT gates. The two qubits are initialized in the \(\Downarrow \Downarrow \rangle \equiv 11\rangle \) state. We observe the Rabi oscillations of both qubits in phase for zCNOT and out of phase for CNOT. At every odd multiple of π/2 rotation of the control qubit the Bell states are created.
Extended Data Fig. 7 Twoqubit GST.
a, Measurement circuit for the twoqubit GST. A modified version of this circuit has been used for Bell state tomography. The green box prepares the qubit 2 in the \(\Uparrow \rangle \) state, then the orange box prepares the qubit 1 in the \(\Uparrow \rangle \) state. The readout step in the blue box (see Methods) determines whether the \(\Downarrow \Downarrow \rangle \) state initialization was successful. Only then the record will be saved. The electron spin is prepared in \(\downarrow \rangle \) during the nuclear spin readout process. Subsequently, the GST sequence is executed. The red box indicates the Q1, Q2 readout step. The total duration of the pulse sequence is 120 ms, of which nuclear spin initialization is 8.6 ms (green and yellow), initial nuclear spin readout is 26.5 ms (blue), 3 ms delay is added for electron initialization (between blue and purple), GST circuit is 10 μs–300 μs (purple), and nuclear readout is 80 ms (orange). b, Measurement results for individual twoqubit GST circuit. The first 145 circuits estimate the preparation and measurement fiducials, and the subsequent circuits are ordered by increasing circuit depth. At the end of a circuit, there are three situations for the target state populations: 1) the population is entirely in one state, while all others are zero; 2) the population is equally spread over two states, while the other two are zero; 3) the population is equally spread over all four states. The measured state populations for the different circuits therefore congregate around the four bands 0, 0.25, 0.5, and 1, as indicated by black dashed lines.
Extended Data Fig. 8 Estimated gate set, from process matrices to error rates.
Experimental GST data were analysed using pyGSTi to obtain selfconsistent maximum likelihood estimates of twoqubit process matrices for all six elementary gates. These are represented (‘Process Matrix’ column) in a gauge that minimizes their average total error, as superoperators in the twoqubit Pauli basis. Green columns indicate positive matrix elements, orange ones are negative. Wireframe sections indicate differences between estimated and ideal (target) process matrices. Those process matrices can be transformed to error generators (‘Error Generator’ column) that isolate those differences, and are zero if the estimated gate equals its target. Each gate’s error generator was decomposed into a sparse sum of Hamiltonian and stochastic elementary error generators^{33}. Those rates are depicted (‘All Error Rates’ column) as contributions to the gate’s total error, with 1σ uncertainties indicated in parentheses. Each nonvanishing elementary error rate (error generators are denoted ‘H’ or ‘S’ followed by a Pauli operator) is listed, and identified with its role in the total error budget (reproduced from Fig. 3). Orange bars indicate stochastic errors, dark blue indicate coherent errors that are intrinsic to the gate, and light blue indicate relational coherent errors that were assigned to this gate. Total height of the blue region indicates the total coherent error, but because coherent error amplitudes add in quadrature, individual components’ heights are proportional to their quadrature.
Extended Data Fig. 9 Simulation of standard and interleaved randomized benchmarking.
All simulated randomized benchmarking experiments used twoqubit Clifford subroutines compiled from the six native gates, requiring (on average) 14.58 individual gate operations per twoqubit Clifford. a, Standard randomized benchmarking, simulated using the GSTestimated gate set, yields a ‘reference’ decay rate of r_{r} = 22.2(2)%, suggesting an average pergate error rate of r_{r}/14.58 ≈ 1.5%. 1σ confidence intervals are indicated in parentheses. b–f, Simulated interleaved randomized benchmarking for the CZ gate, and onequbit X_{π/2} and Y_{π/2} gates on each qubit, yielded interleaved decay rates r_{r} + r_{i}. For each experiment, 1,000 random Clifford sequences were generated, at each of 15 circuit depths m, and simulated using the GST process matrices. Exact probabilities (effectively infinitely many shots of each sequence) were recorded. Inset histograms show the distribution over 1000 random circuits at m = 4. Observed decays are consistent with each gate’s GSTestimated infidelities—for example, 1 − F = 0.79% for the CZ gate (b). Performing these exact randomized benchmarking experiments in the lab would have required running 90,000 circuits to estimate a single parameter (r_{i}) for each gate to the given precision of ±0.25%. Using fewer (<1,000) random circuits at each m would yield lower precision. GST required only 1,500 circuits to estimate all error rates to the same precision.
Supplementary information
Supplementary Information
This file contains supplemental information supporting the main claims of the paper. The information covers the following three components: supplemental data; extended GST analysis; and an analysis of possible causes for the observed twoqubit entangling errors.
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Mądzik, M.T., Asaad, S., Youssry, A. et al. Precision tomography of a threequbit donor quantum processor in silicon. Nature 601, 348–353 (2022). https://doi.org/10.1038/s41586021042927
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DOI: https://doi.org/10.1038/s41586021042927
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