## Abstract

Large-scale quantum computers must be built upon quantum bits that are both highly coherent and locally controllable. We demonstrate the quantum control of the electron and the nuclear spin of a single ^{31}P atom in silicon, using a continuous microwave magnetic field together with nanoscale electrostatic gates. The qubits are tuned into resonance with the microwave field by a local change in electric field, which induces a Stark shift of the qubit energies. This method, known as *A*-gate control, preserves the excellent coherence times and gate fidelities of isolated spins, and can be extended to arbitrarily many qubits without requiring multiple microwave sources.

- Quantum computing
- Single-atom spin qubits
- Silicon nanoelectronics
- Local electrical control
- Magnetic resonance
- Phosphorus donor

The construction of a large-scale quantum computer is among the most exciting scientific challenges of our time. For this purpose, the ability to exploit the advanced fabrication methods developed in semiconductor nanoelectronics (*1*, *2*) would be highly beneficial. Succeeding at this project will depend on the ability to create quantum bits (qubits), which are, at the same time, highly coherent and easy to control and couple to each other. However, quantum coherence and ease of control are often antithetic requirements. Several types of semiconductor spin qubits have been designed to be operated using only electric fields (*3*–*6*). This allows control at the nanometer scale via small electrodes but can lead to unwanted decoherence from charge and gate noise (*7*, *8*). Other spin qubits, such as the electrons and nuclei of ^{31}P donors in silicon (*9*, *10*) or nitrogen-vacancy centers in diamond (*11*, *12*), exhibit outstanding coherence times thanks to their lack of sensitivity to electrical noise and the reduced nuclear spin fluctuations in their vicinity. However, their coherent operation requires high-frequency oscillating magnetic fields, which are difficult to confine to nanometer scales. A solution to reconcile long coherence with local electrical control was already described in the visionary proposal of Kane (*13*) for a donor-based quantum computer in silicon. The resonance frequency of a ^{31}P donor spin depends on the applied magnetic field *B*_{0}, as well as the electron-nuclear hyperfine coupling *A*. The latter can be locally tuned with an electrostatic gate, known as the “*A*-gate,” which deforms the wavefunction of the donor-bound electron and modifies the probability density at the nucleus (*14*). Kane envisaged a multi-qubit quantum computer where a global, always-on microwave magnetic field is by default off-resonance with the qubits. Quantum operations are controlled by locally modifying *A* to bring the desired qubits into resonance with the global microwave field (*15*). This eminently scalable proposal has been further developed to include quantum error correction (*16*). Here, we present the experimental demonstration of local and coherent electrical control of both the electron and the nuclear spin of a single ^{31}P donor in silicon, representing the first realization of Kane’s *A*-gate on a single qubit.

The ^{31}P donor in silicon constitutes a two-qubit system, where both the electron (indicated with |↓〉 or |↑〉) and the nuclear (|⇓〉 or |⇑〉) spin states can be coherently controlled by a magnetic field *B*_{1} oscillating at specific electron spin resonance (ESR) and nuclear magnetic resonance (NMR) frequencies. We fabricated a device that comprises a single ^{31}P donor in an isotopically purified ^{28}Si epilayer (*17*), implanted (*18*) next to the island of a single-electron transistor (SET). The SET is formed under an 8-nm-thick SiO_{2} layer by biasing a set of electrostatic gates (yellow in Fig. 1A). The distance between the donor and SET island is ~20(5) nm, resulting in a tunnel coupling of order 10 kHz. The device is cooled by a dilution refrigerator (electron temperature *T*_{el} ≈ 100 mK) and subjected to a static magnetic field *B*_{0} = 1.55 T applied along the [110] Si crystal axis. Because of the Zeeman effect, the electrochemical potential μ of the donor electron depends on its spin state, with μ_{↑} > μ_{↓}. Another set of gates (pink in Fig. 1A) is used to tune the electrochemical potentials of donor and SET island (μ_{SET}) to the readout position (“Read/Init.” in Fig. 2A), where μ_{↑} > μ_{SET} > μ_{↓} and only the |↑〉 state can tunnel out of the donor. The positive donor charge left behind shifts the SET bias point and causes a current to flow, until a |↓〉 electron tunnels back onto the donor. This spin-dependent tunneling mechanism is therefore used to achieve single-shot electron spin readout (*19*, *20*), as well as |↓〉 initialization. For coherent spin control, an oscillating magnetic field *B*_{1} is delivered to the donor by an on-chip, broadband transmission line terminating in a short-circuited nanoscale antenna (*21*) (blue in Fig. 1A). While manipulating the electron spin state, the gates are tuned such that μ_{↑},_{↓} < μ_{SET} to ensure that the electron cannot escape the donor (“Pulse ESR/NMR” and “*A*-Gate Control” positions in Fig. 2A).

In the present experiment, the same gates used to tune the donor to the readout position are also used to apply an electric field shift, which in turn modifies the spin transition frequencies via the Stark effect. Therefore, we label them *A*-gates henceforth. In Fig. 1B, we illustrate the effect of the electric field on the donor electron. A positive bias on a gate located above the donor pulls the electron wavefunction toward the Si/SiO_{2} interface and away from the nucleus (*14*). This modifies both the hyperfine coupling A (≈117.53 MHz in bulk, and 96.9 MHz for this device) (*14*, *22*) and the electron gyromagnetic ratio γ_{e} (≈27.97 GHz/T in bulk) (*23*). The ESR frequencies ν_{e1,2} depend on both these parameters and on the state of the ^{31}P nuclear spin. In the limit γ_{e}*B*_{0} ≫ *A*, we find ν_{e1} = γ_{e}*B*_{0} − *A*/2 (for nuclear spin |⇓〉) and ν_{e2} = γ_{e}*B*_{0} + *A*/2 (for |⇑〉), as shown in Fig. 1C. Identifying whether the instantaneous ESR frequency is ν_{e1} or ν_{e2} constitutes a single-shot, quantum nondemolition nuclear spin readout (*10*). The NMR transitions can be gate-tuned as well, albeit only via the Stark shift of *A*, because ν_{n1} = *A*/2 + γ_{n}*B*_{0} (for electron spin |↓〉) and ν_{n2} = *A*/2 − γ_{n}*B*_{0} (for |↑〉), where γ_{n} = 17.23 MHz/T is the ^{31}P nuclear gyromagnetic ratio.

During pulse ESR/NMR experiments, we normally operate the electrostatic gates in a compensated manner to keep μ_{SET} constant while shifting μ_{↑,↓} with respect to it (*19*) (“Pulse ESR/NMR” position in Fig. 2A). This, however, results in a limited variation of the electric field at the donor site. To induce a significant Stark shift, we adopted an uncompensated gating scheme, where both μ_{↑},_{↓} and μ_{SET} are drastically lowered (“*A*-Gate Control” position in Fig. 2A; see section S4 for details on the exact gate configuration). The resulting change in electric field can be calculated with a finite element Poisson equation solver (TCAD) for the specific device geometry and the triangulated donor location (see sections S5 and S6 for details) (*24*).

We demonstrate the gate-induced shift of the ESR frequencies by performing, with conventional pulse ESR, a series of Ramsey experiments on the electron spin at different values of *V*_{A} (Fig. 2B), where we define *V*_{A} = 0 as the readout position. The spin is rotated from |↓〉 to the *xy* plane by a π/2 pulse at frequency ν_{MW}, left to freely precess for a time τ, and then rotated by π/2 again. The accumulation of a phase shift between the spin precession at ν_{e}(*V*_{A}) and the MW reference clock at ν_{MW} gives rise to oscillations in the probability of finding the electron |↑〉 at the end of the sequence. The frequency of the Ramsey fringes gives a very accurate value for ν_{e}(*V*_{A}) − ν_{MW}.

Both ν_{e1} and ν_{e2} shift to lower frequencies upon increasing *V*_{A} (Fig. 2B). This indicates that a significant Stark shift of γ_{e} (that is, the electron *g*-factor) is taking place, in addition to the *A*-shift. Linear fits to ν_{e1,2}(*V*_{A}) yield slopes *d*ν_{e1}/*dV*_{A} = −2.27(6) MHz/V and *d*ν_{e2}/*dV*_{A} = −1.36(3) MHz/V. Using the expressions given in Fig. 2B, we extract the tuning parameters α_{A} = *d*_{A}/*dV*_{A} = *d*ν_{e2}/*dV*_{A} − *d*ν_{e1}/*dV*_{A} = 0.91(7) MHz/V and α_{γe}*B*_{0} = *d*γ_{e}*B*_{0}/*dV*_{A} = (*d*ν_{e2}/*dV*_{A} + *d*ν_{e1}/*dV*_{A})/2 = –1.81(5) MHz/V, with α_{γe} = *d*γ_{e}/*dV*_{A} = −1.17(3) MHz/V/T at *B*_{0} = 1.55 T. The positive value of α_{A} indicates that increasing *V*_{A} leads to an increase in the electron probability density at the nucleus. This is because a strong electric field is already present for the purpose of forming the SET, whereas increasing *V*_{A} causes an additional electric field in the opposite direction, thus an overall reduction of the hyperfine Stark shift (see finite element simulations in section S6). The absolute value of the hyperfine coupling and its tunability is in agreement with atomistic tight-binding simulations with NEMO-3D (*25*, *26*) for the range of donor positions, electric fields, and strain expected for this device (see section S7).

An *A*-gate voltage *V*_{A} ~300 meV results in a frequency tuning range = 400 to 700 kHz. Thanks to the use of an isotopically enriched ^{28}Si epilayer (*17*), this is a factor >200 larger than the intrinsic linewidth = 1.8 kHz of the ESR transitions (*22*). For the same *V*_{A}, the NMR frequencies can be shifted by = 125 to 150 kHz by the Stark shift of *A* alone, which is a factor ~250 higher than the intrinsic linewidth = 0.5 kHz (*22*).

Having calibrated the voltage-controlled qubit frequency shifts, we demonstrate how to use *A*-gate pulses to perform coherent control of the qubit states around the Bloch sphere in the presence of a continuous-wave (CW) oscillating magnetic field *B*_{1}. We demonstrate this on both the electron and the nuclear spin qubits. Figure 3A is a schematic of the electrically controlled Rabi sequence. We pulse *V*_{A} to *V*_{r}, the voltage needed to tune the spin transition in resonance with the MW or RF source, for a time *t*_{p}, to coherently drive the spin around the *x* axis of the Bloch sphere, defined in the reference frame rotating at ν_{MW} (for ESR) or ν_{RF} (for NMR). The coherent Rabi oscillations (Fig. 3, B and C) are used to calibrate the duration of the control pulses for any desired rotation angle.

An electrically controlled Ramsey experiment (Fig. 3D) is obtained by tuning the spin transition frequencies into resonance for the duration of a π/2 rotation, then moving them to a detuned value ν_{det} for a wait time τ to accumulate a phase shift Φ_{R} = τ(ν(*V*_{r}) − ν_{det}) with respect to the ν(*V*_{r}) reference frame, and finally tuning them back into resonance for the second π/2 rotation (Fig. 3, E and F). Qubit rotations around the *y* axis can be achieved by accumulating an additional Φ_{R} = π/2 before bringing the spin into resonance. This allows for full two-axis (*x* and *y*) control of the qubits’ states on the Bloch sphere. Alternatively, we can produce a *Z*-gate by controlling the phase Φ_{R} that the qubit accumulates while off-resonance.

We measure the qubits’ coherence times *T*_{2} by performing Hahn echo and Carr-Purcell-Meiboom-Gill (CPMG) dynamical decoupling sequences (Fig. 3, G to K). For these measurements, the wait times τ are chosen such that they always result in a multiple of 2π phase shift. The Hahn echo yields = 0.97(6) ms and = 5.1(3) ms for the donor electron and nuclear spins, respectively (Fig. 3, H and I). A CPMG sequence (see schematic in Fig. 3G for details) further decouples the qubits from low-frequency noise and extends the coherence times to up to =10.0(8) ms and = 98(9) ms by applying 32 and 256 π pulses, respectively (Fig. 3, J and K). The coherence times of the electrically controlled electron qubit, measured in milliseconds, can be fitted by (*N*) = 1.03*N*^{0.67}. Conventional pulse ESR operation (black diamonds in Fig. 3J) gives nearly identical values, = 0.93*N*^{0.70} (see also section S10 for more data). The electrical control method preserves the excellent spin coherence of the qubits, because no additional coupling between the spin states and the environment has been introduced.

We apply the two-axis control to construct the complete set of 1-qubit Clifford gates, and conduct randomized benchmarking (*27*, *28*) experiments (Fig. 4, A and B) to quantify the average gate fidelities using electrically controlled qubit manipulation (Fig. 4). The method relies upon the measurement of the probability (*N*) of arriving at the correct final qubit state after performing a long sequence of randomly chosen quantum gates. In the presence of gate errors, the probability decays according to the number of operations *N*. We then extract the average fidelity for a single Clifford gate ^{c} by fitting the equation1to the data (*28*). *M* is a free parameter and depends on the initialization and readout fidelity. We obtain = 99.0(1)% and = 99.3(1)% for the electron and the nuclear spin, respectively. Because each Clifford gate is composed on average of 1.875 individual gate operations (*28*, *29*), we can also quote average single-gate fidelities of = 99.4(1)% for the electron and = 99.6(1)% for the nucleus. These gate fidelities are comparable to those obtained by pulse ESR/NMR randomized benchmarking for similar microwave powers (*28*) and are mostly limited by the ratio of gate time to coherence time.

The speed of an electrically controlled gate operation is inherently limited by the voltage tunability of the resonance frequencies. The excitation profile of the CW field (Fig. 3A) must be narrow enough to leave the qubits unperturbed while off-resonance, imposing the condition γ*B*_{1} ≪ Δν. For linearly oscillating *B*_{1}, (γ*B*_{1})^{−1} is the duration of a π rotation. Therefore, Δν < 1 MHz in this ^{31}P device requires gate times >10 μs.

The qubit control method demonstrated here is applicable to any resonantly driven qubit, where the resonance frequency ν_{r} can be quickly and locally controlled by an electric field, and shifted by much more than the resonance linewidth. A wide variety of qubits can potentially fulfill this condition. For spins in diamond (*30*) and silicon carbide (*31*), ν_{r} can be tuned by modifying the crystal field parameters in the spin Hamiltonian. Magnetic molecules can have tunable ν_{r} through a hyperfine Stark effect (*32*) similar to the one shown here. Several types of semiconductor quantum dot qubits exhibit tunable electron spin *g*-factor (*33*, *34*) through spin-orbit coupling effects, or tunable splitting through the interplay of valley-orbit and tunnel couplings (*6*).

Because of the high cost of vector microwave signal generators, it seems implausible that future multi-qubit experiments will resort to a dedicated source for each qubit. Time- and frequency-multiplexing qubit control is possible in proof-of-principle experiments but is impractical in large fault-tolerant quantum processors. In most error correction schemes, fault tolerance is only guaranteed if all qubits can be operated simultaneously at any time. The method of qubit control demonstrated here fulfills all the practical requirements for a large-scale quantum computer, because control gates can be applied simultaneously to arbitrarily many qubits while requiring only one CW microwave source together with inexpensive multichannel baseband pulse generators.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/1/3/e1500022/DC1

Fig. S1. Frequency response.

Fig. S2. Donor triangulation.

Fig. S3. Electric field simulations.

Fig. S4. Electric field simulations.

Fig. S5. Electric field simulations.

Fig. S6. Atomistic tight binding simulations of the hyperfine coupling for a donor at the location determined in section S5, for the electric fields simulated in section S6, and subject to lattice strain.

Fig. S7. Electrically controlled ESR spectrum.

Fig. S8. Time evolution simulations of the electrically controlled ESR spectrum.

Fig. S9. Electrically controlled Rabi spectrum.

Fig. S10. Electrically controlled electron Ramseys.

Fig. S11. Electrically controlled nuclear Ramseys.

Fig. S12. Electrically controlled electron coherence times.

Fig. S13. Electrically controlled nuclear coherence times.

Table S1. Relative gate capacitances used for triangulation of the donor position.

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## REFERENCES AND NOTES

**Acknowledgments:**We thank S. Simmons, G. Tosi, and C. D. Hill for fruitful discussions and comments. This research was funded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027) and the U.S. Army Research Office (W911NF-13-1-0024). We acknowledge support from the Australian National Fabrication Facility and from the laboratory of R. Elliman at the Australian National University for the ion implantation facilities. The work at Keio has been supported in part by FIRST, the Core-to-Core Program by the Japan Society for the Promotion of Science, and the Grant-in-Aid for Scientific Research and Project for Developing Innovation Systems by MEXT. NCN/nanohub.org computational resources funded by the National Science Foundation under contract number EEC-1227110 were used in this work.

**Competing interests:**The authors declare that they have no competing interests.

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