Abstract
Practical quantum computers require a large network of highly coherent qubits, interconnected in a design robust against errors. Donor spins in silicon provide stateoftheart coherence and quantum gate fidelities, in a platform adapted from industrial semiconductor processing. Here we present a scalable design for a silicon quantum processor that does not require precise donor placement and leaves ample space for the routing of interconnects and readout devices. We introduce the flipflop qubit, a combination of the electronnuclear spin states of a phosphorus donor that can be controlled by microwave electric fields. Twoqubit gates exploit a secondorder electric dipoledipole interaction, allowing selective coupling beyond the nearestneighbor, at separations of hundreds of nanometers, while microwave resonators can extend the entanglement to macroscopic distances. We predict gate fidelities within faulttolerance thresholds using realistic noise models. This design provides a realizable blueprint for scalable spinbased quantum computers in silicon.
Introduction
The successful implementation of quantum algorithms requires incorporation of error correction codes^{1} that deal with the fragile nature of qubits. The highest tolerances in error rates are found when using nearestneighbor topological codes^{2}, longdistance entanglement links^{3}, or a combination of both^{4}. There exist several physical platforms where state preservation^{5,6,7}, qubit control^{8,9,10,11}, and twoqubit logic gates^{8, 12} are achieved with faulttolerant fidelities. The ultimate goal is to integrate a large number of qubits in expandable arrays to construct a scalable, universal quantum processor.
Donor spin qubits in silicon are an appealing physical platform for that goal, due to their integrability with metaloxidesemiconductor (MOS) structure and nanometric unit size^{13}. By using isotopically enriched ^{28}Si as the substrate material^{14}, donor spins offer coherence times around a second (for the electron) or a minute (for the nucleus)^{7}, up to hours in bulk ensembles^{6}, and control error rates as small as 10^{−4} ^{11}. However, integrating several of these qubits in a scalable architecture remains a formidable challenge, mainly because of the difficulty in achieving reliable twoqubit gates.
The seminal Kane proposal^{15} for a nuclearspin quantum computer in silicon described the use of shortrange exchange interactions J between donorbound electrons, to mediate an effective internuclear coupling of order ~100 kHz at a ~15 nm distance. However, the exchange interaction has an exponential and oscillatory spatial behavior that can result in an order of magnitude variation in strength upon displacement by a single lattice site^{16, 17}. Notwithstanding, plenty of progress has been made in the experimental demonstration of the building blocks of a Kanetype processor^{18,19,20,21}, including the observation of interdonor exchange^{22,23,24}. Slightly relaxed requirements on donor placement can be found when using a hyperfinecontrolled exchange interaction between electron spin qubits^{25}, or a slower magnetic dipoledipole coupling effective at ~30 nm distances^{26}. Other proposals space donors further apart by introducing some intermediate coupler, e.g., donor chains^{27, 28}, chargecoupled devices^{29}, ferromagnets^{30}, probe spins^{31}, or quantum dots^{32}.
Here we introduce the design of a largescale, donorbased silicon quantum processor based upon electric dipole interactions. This processor could be fabricated using existing technology, since it does not require precise donor placement. The large interqubit spacing, >150 nm, leaves sufficient space to intersperse classical control and readout devices, while retaining some of the compactness of atomicsize qubits. Stabilization schemes largely decouple the qubits from electric noise while still keeping them sensitive to electric drive and mutual coupling. Finally, the whole structure retains the standard silicon MOS materials stack, important for ultimate manufacturability.
Results
Coupling Si:P spin qubits to electric fields
The phosphorus donor in silicon comprises an electron spin S = 1/2 with gyromagnetic ratio γ _{e} = 27.97 GHz T^{−1} and basis states \(\left \downarrow \right\rangle \), \(\left \uparrow \right\rangle \), and a nuclear spin I = 1/2 with gyromagnetic ratio γ _{n} = 17.23 MHz T^{−1} and basis states \(\left \Downarrow \right\rangle ,\left \Uparrow \right\rangle \). The electron interacts with the nucleus through the hyperfine coupling A ≈ 117 MHz. When placed in a large magnetic field B _{0} (\({\gamma _ + }{B_0} \gg A\), with γ _{+} = γ _{e} + γ _{n}), the eigenstates of the system are the separable tensor products of the basis states, i.e., \(\left { \downarrow \Uparrow } \right\rangle \), \(\left { \downarrow \Downarrow } \right\rangle \), \(\left { \uparrow \Downarrow } \right\rangle \), \(\left { \uparrow \Uparrow } \right\rangle \) (Fig. 1c). The electron and the nucleus can be operated as single qubits by applying oscillating magnetic fields resonant with any of the transitions frequencies between eigenstates that differ by the flipping of one of the spins, e.g., \(\left { \downarrow \Uparrow } \right\rangle \) ↔ \(\left { \uparrow \Uparrow } \right\rangle \) for the electron qubit, etc. (Fig. 1c).
We envisage a device where a shallow ^{31}P donor is embedded in an isotopically enriched ^{28}Si crystal at a depth z _{d} from the interface with a thin SiO_{2} layer (Fig. 1a). The orbital wavefunction ψ of the donorbound electron can be controlled by a vertical electric field E _{ z } applied by a metal gate on top. It changes from a bulklike donor state at low electric fields to an interfacelike state at high fields^{33, 34} (insets in Fig. 1d). The hyperfine interaction A(E _{ z }), proportional to the square amplitude of the electron wavefunction at the donor site ψ(0, 0, z _{d})^{2}, changes accordingly from the bulk value A ≈ 117 MHz to A ≈ 0 when the electron is fully displaced to the interface (Fig. 1d). Shifting the electron wavefunction also results in the creation of an electric dipole μ _{e} = ed, where e is the electron charge and d is the separation between the mean positions of the donorbound and interfacebound wavefunctions (d ≲ z _{d}, see Supplementary Note 1). The induced electric dipole μ _{e} has been largely overlooked in the past, but plays a crucial role in this proposal.
The key idea is to define a new qubit, called henceforth the flipflop qubit, described in the subspace spanned by the states \(\left { \downarrow \Uparrow } \right\rangle \), \(\left { \uparrow \Downarrow } \right\rangle \). Transitions between these basis states cannot be induced by magnetic resonance, because there is no change in the zcomponent of the total angular momentum. However, the hyperfine interaction, A S ⋅ I, is a transverse term in the flipflop basis, since its eigenstates are \(S = \left( {\left { \downarrow \Uparrow } \right\rangle  \left { \uparrow \Downarrow } \right\rangle } \right){\rm{/}}\sqrt 2 \) and \({T_0} = \left( {\left { \downarrow \Uparrow } \right\rangle + \left { \uparrow \Downarrow } \right\rangle } \right){\rm{/}}\sqrt 2 \) (Fig. 1b). Therefore, electrically modulating A(E _{z}) at the frequency
corresponding to the flipflop qubit energy splitting, causes an electric dipole spin resonance (EDSR) transition between the \(\left { \downarrow \Uparrow } \right\rangle \), \(\left { \uparrow \Downarrow } \right\rangle \) basis states^{35, 36} (Fig. 1c). This transition is faster at the “ionization point”, where the electron is shared halfway between donor and interface, since A(E _{ z }) can vary strongly upon the application of a small voltage on the top gate.
Electrical noise and relaxation
Since the qubit operation is based upon the use of electric fields, a natural concern is the fragility of the qubit states in the presence of electric noise. Below we show that there are special bias points that render the flipflop qubit operation highly robust against noise.
A quantummechanical description of the system is obtained by treating also the electron position as a twolevel system (effectively a charge qubit; see Supplementary Note 1 for a justification of this twolevel approximation), where the vertical position of the electron is represented by a Pauli σ _{ z } operator, with eigenvectors d 〉, for the electron at the donor, and i 〉 at the interface (Fig. 1a, d). The simplified orbital Hamiltonian reads (in units of Hz):
where V _{t} is the tunnel coupling between the donor and the interface potential wells, \(E_{z}^0\) is the vertical electric field at the ionization point, and h is the Planck constant. The electron ground g 〉 and excited e 〉 orbital eigenstates depend on E _{ z } (Fig. 1d) and have an energy difference given by:
At the ionization point, the energy difference between eigenstates \(\left e \right\rangle = \left( {\left d \right\rangle + \left i \right\rangle } \right){\rm{/}}\sqrt 2 \) and \(\left g \right\rangle = \left( {\left d \right\rangle  \left i \right\rangle } \right)/\sqrt 2 \) is minimum and equal to V _{t} (Fig. 2a), and therefore firstorder insensitive to electric noise, ∂\({\epsilon _o}\)/∂E _{z} = 0. This bias point is referred to as the “charge qubit sweet spot”^{37} (CQSS—Fig. 2a).
Conversely, the bare flipflop qubit energy is expected to depend strongly on E _{ z }, through the combined effect of the hyperfine interaction A (Eq. 1) and the orbital dependence of the electron gyromagnetic ratio, γ _{e}. Indeed, the gyromagnetic ratio of an electron confined at a Si/SiO_{2} interface can differ from that of a donorbound electron by a relative amount Δ_{ γ } up to 0.7%^{38}. Therefore, the Zeeman terms in the Hamiltonian must include a dependence of the electron Zeeman splitting on its orbital position, i.e., the charge qubit σ _{ z } operator:
We can also write the hyperfine coupling as an operator that depends on the charge qubit state:
Indeed, this simple twolevel approximation, shown as a black line in Fig. 1d, reproduces the full tightbiding simulations (yellow dots).
The overall flipflop qubit transition frequency as a function of E _{ z } becomes:
shown in Fig. 2a (dashed line), where we assumed Δ_{ γ } = −0.2%^{38}. \({\epsilon _{{\rm{ff}}}}\)(A, γ _{e}) shows a steep slope around the ionization point, mostly caused by the E _{ z }dependence of γ _{e} (the dependence on A is less significant because \({\gamma _ + }{B_0} \gg A\)). Therefore, while \({E_{z}} \approx E_{z}^0\) is the fastest operation point for the flipflop qubit driven by a resonant modulation of A, one might expect it to be the most prone to qubit dephasing from charge and gate noise, through the influence of E _{ z } on γ _{e}.
However, computing instead the full flipflop qubit Hamiltonian,
reveals that the qubit transition frequency has an extra bend around the ionization point (Fig. 2a, thick yellow line). This comes from Eq. (5), which provides a transverse coupling g _{so} between the flipflop and charge qubits (inset in Fig. 2a):
As a result, the electron orbit dispersively shifts the flipflop qubit by, to second order:
where δ _{so} = \({\epsilon _o}\) − \({\epsilon _{{\rm{ff}}}}\), reducing the flipflop qubit frequency to:
D _{orb}(E _{ z }) is largest around \({E_{z}} \approx E_{z}^0\), since δ _{so} is lowest (i.e., the charge qubit frequency comes closest to the flipflop qubit, Fig. 2a) and g _{so} is highest. Equation (10) (thin black line in Fig. 2a) agrees with full numerical simulations of the Hamiltonian in Eq. (7).
Such a dispersive shift stabilizes the flipflop precession frequency against noise. To quantify that, we assume a quasistatic electric field noise with 100 V m^{−1} r.m.s. amplitude along the donordot direction (zaxis in Fig. 1a). This noise is equivalent to a 1.5 μeV charge detuning noise for d = 15 nm, consistent with experimentally observed values in similar silicon devices^{39,40,41}—see Supplementary Note 3. The estimated—see Methods section—dephasing rates can be as low as \(1{\rm{/}}T_2^ * \approx 3\) kHz (Fig. 2b), comparable to the ones due to magnetic noise (\(1/T_2^ * \approx 1\) kHz in ^{28}Si nanostructures^{7}). This can be understood from Fig. 2c, which shows the qubit precession frequency dependence on E _{ z }, for three different values of V _{t}. For small detunings δ _{so}, i.e., V _{t} close to \({\epsilon _{{\rm{ff}}}}\), the dispersive shift around the ionization point is strong, yielding two firstorder “clock transitions” (CT), where ∂\({\epsilon _{{\rm{ff}}}}\)/∂E _{ z } = 0 where the dephasing rate is reduced. By increasing V _{t}, the two firstorder points merge into a single one in which both the first and second derivatives vanish, yielding the slowest qubit dephasing.
Another source of errors could come from relaxation via coupling to phonons. This is not an issue for bulk donors, where electron spin relaxation time is \({T_{1,{\rm{s}}}} \gg 1\) s^{18}. However, due to the particular valley composition of the flipflop qubit near the ionization point, its relaxation rate 1/T _{1,ff} due to chargephonon coupling is enhanced^{42}. We estimate it by noting that, if \({\delta _{{\rm{so}}}} \gg {g_{{\rm{so}}}}\), 1/T _{1,ff} is equal to the amount of charge excited state in the flipflop eigenstates^{43} times the charge relaxation rate^{42}:
where T _{1,o} is the charge qubit lifetime and Θ ≈ 2.37 × 10^{−24} s^{2} is determined by the silicon crystal properties^{42}. Therefore, as can be seen from Fig. 2d, the higher the detuning δ _{so}, the slower the relaxation. In particular, at the secondorder CT, the qubit dephasing can be limited by relaxation, \(1{\rm{/}}T_2^* = 1{\rm{/}}2{T_1} \approx {10^4}\) Hz. This limitation can be overcome by reducing B _{0} (Fig. 2e).
Tuning a flipflop qubit into a clock transition requires the ability to tune the tunnel coupling V _{t}. The latter is difficult to control at the fabrication stage, given its exponential dependence on donor depth, together with oscillations at the atomic scale^{44} arising from a similar valley interference effect as the one afflicting the exchange interaction^{16}. Indeed, ionimplanting a donor at z _{d} ≈ 15 nm below the interface happens with a vertical uncertainty of order ±10nm^{45}, resulting in more than two orders of magnitude uncertainty in V _{t} ^{44}. Therefore, it is crucial to implement a method to tune V _{t} in situ. A possible solution is to displace the location of the interface wavefunction laterally, which in turn modifies the overlap between the donor and interface wavefunctions and therefore V _{t}. This can be done by adding two gates on either side of the top gate, which pulls the donor electron to the interface (Fig. 2f), in such a way that a relative voltage between the gates can modify the interface lateral potential landscape. This gate stack is identical to the wellestablished scheme for the confinement of single electrons in Si quantum dots^{10}. This technique allows V _{t} to be tuned by at least wo orders of magnitude (Fig. 2g), therefore circumventing the uncertainty in donor depth and V _{t} arising from ionimplantation.
Adiabatic phase control
The presence of slow dephasing regions is important to control the qubit phase with high fidelity. In our quantum processor, idle qubits are decoupled from electric fields by fully displacing the electron either to the interface or to the donor. Performing quantum operations on the qubit requires displacing the electrons close to the ionization point, which in turn changes its precession frequency (Fig. 2a). As a result, the accumulated phase must be corrected after quantum operations. This is optimally done by moving the electron to the secondorder clock transition, therefore minimizing dephasing errors. At this point, the flipflop qubit phase precesses \({\rm{\∼}}{\Delta _\gamma }{\gamma _{\rm{e}}}{B_0}{\rm{/}}2  {D_{{\rm{orb}}}}\) faster than its idle point, and therefore any phase correction in a 2π period can be applied within tens of ns. The dephasing rate at the CT, on the order of a few kHz, would cause very small errors (<10^{−4}). However, while moving the electron from the interface toward the donor, the flipflop qubit goes through regions of fast dephasing (Fig. 2b), and therefore this operation has to be performed as quickly as possible. It also has to be slow enough as to avoid erros due to nonadiabaticity, which include, e.g., leakage to unwanted highenergy states. These errors depend on the adiabatic factor K, which quantifies the fractional rate of change of the system’s eigenstates (the higher the value of K, the more adiabatic and slower is the process—see Methods section).
In Fig. 3a, we plot the time dynamics of an initial state \(\left g \right\rangle \otimes \left( {\left { \downarrow \Uparrow } \right\rangle + \left { \uparrow \Downarrow } \right\rangle } \right){\rm{/}}\sqrt 2 \) while sweeping E _{ z } adiabatically (K = 50) to move the electron from the interface to the secondorder CT and back, in order to realize a π zgate. The initial adiabatic setup part consists of a fast sweep (0.8 ns), allowed by the large charge qubit splitting when \({E_{\rm{z}}} \gg E_{\rm{z}}^0\), followed by a slower sweep (3.5 ns), limited by the proximity of excited charge states to the flipflop qubit when \({E_{\rm{z}}} \approx E_{\rm{z}}^0\). The electron then remains at the CT for 60 ns, before adiabatically moving back to the interface. During the total 69 ns, the flipflop qubit phase is shifted by π, with adiabatic errors, averaged over a set of initial flipflop states—see Methods section—around 10^{−4}. These errors can be controlled with the factor K, which determines the setup time (see Fig. 3b).
Quasistatic E _{ z } noise can increase errors, due to dephasing (Fig. 3c). At realistic noise levels (100 V m^{−1}), the gate error rate is found to be <10^{−4}. Similar error levels arise due to relaxation, which remains below 3 × 10^{4} Hz (Fig. 2d).
Note that the presence of clock transitions does not affect the ability to use E _{ac} to resonantly drive the qubit, since the transverse term A(E _{ z }) still responds fully to the electric field (this is similar to the case of magnetic clock transitions, e.g,. in Si:Bi^{46}).
Electric drive of the flipflop qubit
We now explain how highfidelity onequbit x(y)gates can be achieved via electric drive of the flipflop qubit. The fastest onequbit gates are obtained when the electron is around the ionization point, where ∂A/∂E _{z} is maximum (Fig. 1d). A vertical oscillating electric field of amplitude E _{ac} is applied (Fig. 4a) in resonance with the flipflop qubit, i.e., ν _{E} = \({\epsilon _{{\rm{ff}}}}\). A large detuning \({\delta _{{\rm{so}}}} \gg {g_{{\rm{so}}}}\) (Fig. 4b) ensures the least amount of the charge excited state \(\left e \right\rangle \) in the qubit eigenstates, minimizing qubit relaxation via chargephonon coupling. The flipflop qubit is still driven, via a secondorder process, at a rate (halfRabi frequency):
where δ _{E} = ν _{E} − \({\epsilon _o}\) and g _{E} is the driven electric coupling rate between the two charge eigenstates:
where E _{ac} is the amplitude of a sinusoidal drive. Equation (13) provides another explanation of why the fastest onequbit gates are obtained when the electron is at the ionization point: δ _{so} and δ _{E} are minimum (\({\epsilon _o}\) is minimum), and g _{so} and g _{E} are maximum (Eqs. (8) and (14)).
The electrical drive can cause some excitation of the charge qubit. It is therefore convenient to turn E _{ac} on/off adiabatically to make sure the charge is deexcited at the end of the gate. Figure 4c shows the E _{ac} time evolution needed for a π/2 xgate, where we have assumed an adiabatic factor K = 30, sufficient for leakage errors <10^{−3}. E _{ac} increases steadily until a π/4 rotation is completed, after which E _{ac} is gradually switched off to achieve an adiabatic π/2 xgate. An average 4% excitation of the charge qubit causes a ~4 × 10^{4} Hz relaxation rate of the encoded quantum state (Eq. 12), or error levels close to 10^{−3}.
We then investigate how the total π/2 xgate errors depend on the biasing of the electron wavefunction. At the ionization point, \(E_{z} = E_{z}^0\), error levels close to 10^{−3} are found over a wide range of V _{t} (Fig. 4e). The K = 30 choice ensures adiabatic errors <10^{−3} with an oscillatory character typical of adiabatic processes^{47}. At small V _{t} (and therefore small detuning δ _{so}), the qubit eigenstates contain a substantial amount of charge, causing more errors due to chargephonon relaxation. Increasing the detuning δ _{E} with larger V _{t} allows for a faster adiabatic sweep and higher powers (Fig. 4d), yielding shorter gate times and therefore less errors due to quasistatic noise. Still, the incident power is at least three orders of magnitude lower than the one needed to drive donor electron spin qubits, at the same Rabi frequency, with oscillating magnetic fields^{7, 19}.
As Fig. 4f shows, low error rates are still available away from the ionization point, even though best values are found at \({E_{z}} = E_{z}^0\). This is because our gate times are so fast that dephasing, and therefore CTs, do not play a crucial role. Instead, quasistatic E _{z} noise cause errors mainly by modulating the driving strength \(g_{\rm{E}}^{{\rm{ff}}}\), causing “gate time jitter”. Indeed, the gate time is sensitive to the orbital transition frequency \({\epsilon _o}\) (Eq. 13), and therefore gate errors are minimized close to the charge qubit sweet spot (CQSS), where ∂\({\epsilon _o}\)/∂E _{ z } = 0 (Fig. 2a).
Finally, as Fig. 4g shows, lower quasistatic E _{z} noise can cause less errors, provided that the adiabatic factor K is increased, to reduce leakage errors, up to an optimum value where gate times are still fast as to keep noise errors low. Relaxation errors could also be reduced by reducing B _{0} (recall Fig. 2e).
A number of other noise sources, including high frequency charge noise, JohnsonNyquist, and evanescentwave Johnson noise^{48} (EWJN) also affect qubits that are sensitive to electric fields. However, as we discuss in Supplementary Note 3, the corresponding error rates are much lower than the ones already previously mentioned—see all estimated error levels in Table 1.
Twoqubit coupling via electric dipole interaction
We now present the method to couple donor spins that lies at the heart of our scalable quantum processor. It exploits the electric dipole that naturally arises when a donorelectron wavefunction is biased to the ionization point (Fig. 5a), due to the fact that a negative charge has been partly displaced away from the positive ^{31}P nucleus. The electric field produced by this induced dipole in turn, modifies the energy of a nearby donor which is also biased at the ionization point, resulting in a longrange coupling between the two.
The interaction energy between two distant dipoles, μ _{1} and μ _{2}, oriented perpendicularly to their separation, r, is^{49} \({V_{{\rm{dip}}}} = {\mu _1}{\mu _2}{\rm{/}}\left( {4\pi {\varepsilon _{\rm{r}}}{\varepsilon _0}{r^3}} \right)\), where ε _{0} is the vacuum permittivity and \(\epsilon \) _{r} the material’s dielectric constant (ε _{r} = 11.7 in silicon). The electric dipole of each donorinterface state is μ _{ i } = ed _{i}(1 + σ _{ z,i})/2, implying that the dipoledipole interaction Hamiltonian is:
This electric dipoledipole interaction is therefore equivalent to a small shift in the equilibrium orbital position of both electrons plus a coupling term between the charge qubits (blue dashed rectangle in Fig. 5b) equal to:
Note that this interaction can be stronger due to the presence of a metallic interface on top of the qubits, which enhances vertical dipoles—see Supplementary Note 2. Most importantly, since each flipflop qubit is coupled to their electron position (Eq. 5), the electric dipoledipole interaction provides a natural way to couple two distant flipflop qubits.
Indeed, the effective coupling rate between two flipflop qubits at the ionization point, Fig. 5d, exceeds 10 MHz around two narrow regions. These bands can be understood from the energylevel diagram shown in Fig. 5c. The two charge qubits in Fig. 5b form hybridized molecular states, which are coupled to each flipflop qubit. The twoqubit coupling rate is maximum when in resonance with a molecular state. However, this regime induces too many relaxation errors due to resonant charge excitation. Therefore, it is best to detune the flipflop qubits from the molecular states, while still keeping a substantial interqubit coupling rate, via a secondorder process, equal to:
where D _{dd} is the charge eigenenergies shift and α, β the eigenstates coefficients—see Fig. 5c caption.
Twoqubit gates start with both electrons at the interface, where qubits are decoupled since the electric dipoles and the hyperfine interactions are firstorder insensitive to vertical electric fields. Indeed, from Eq. (18), \(g_{{\rm{2q}}}^{{\rm{ff}}}\) is negligible since g _{so} vanishes and δ _{so} diverges. The electrons are then simultaneously and adiabatically displaced to the ionization point for a time necessary for an \(\sqrt {i{\rm{SWAP}}} \) gate, before returning to the interface. In Fig. 6a, we show the dynamics of a twoqubit gate performed with an adiabatic factor K = 30, following the trajectory shown in Fig. 5e. Similarly to onequbit zgates, the electron is first displaced in a fast time scale (~0.3 ns) set by the charge qubit parameters (\({\epsilon _0}\) and V _{t}), followed by a slower sweep (~19 ns) set by the spincharge coupling parameters (δ _{so} and g _{so}), until it reaches the ionization point. The electron remains still for a short time before the whole process is then reversed. In the end, a \(\sqrt {i{\rm{SWAP}}} \) gate is performed. While some amount of charge is excited during the process, it goes back to its ground state, \(\left {gg} \right\rangle \), with an adiabatic error around 10^{−3}.
We quantify the twoqubit gate fidelity in presence of the most deleterious noise types for our qubits, namely quasistatic E _{z} noise and chargephonon relaxation. For this, we observe that the optimal gate fidelities are achieved when \({E_{z}}\left( {{\tau _{\sqrt {i{\rm{SWAP}}} }}{\rm{/}}2} \right) \approx E_{z}^0\). Similarly to onequbit xgates, this happens because \(\sqrt {i{\rm{SWAP}}} \) gates are sensitive to gate time jitter, and therefore errors are minimized at the CQSS, where \(g_{{\rm{2q}}}^{{\rm{ff}}}\) is robust against E _{z} noise to first order—recall Fig. 5e and Eq. (18). An optimization algorithm finds the best adiabatic factor K that minimizes errors due to E _{z} noise for each value of V _{t,1} = V _{t,2} = V _{t}. The result is shown in Fig. 6b. Smaller detunings δ _{so} (small V _{t}) result in shorter gate times, which in turn reduces errors from quasistatic noise. However, this also implies a larger admixture of charge in the qubit eigenstates, which slightly increases relaxation errors. The lowest error rates, ~3 × 10^{−3} are found at small detunings, V _{t} − \({\epsilon _{{\rm{ff}}}}\) − g _{dd} ≈ 100 MHz (V _{t} ≈ 11.59 GHz). At even smaller detunings, the twoqubit coupling rate becomes too fast, requiring faster adiabatic sweeps to avoid overrotation (lower K, Fig. 6b) and generating more leakage errors. The gate errors remain within 10^{−3} − 10^{−2} for a wide range of V _{t}. Finally, we estimate in Fig. 6c how noise errors depend on the noise amplitude and adiabatic factor K, which sets the gate time.
Our proposed twoqubit gates are not only well protected against noise, but also robust against donor misplacement. Variations in r, d _{1}, and d _{2} mainly cause variations in the charge qubits coupling g _{dd}, therefore simply changing the energy separation between molecular charge states (Fig. 5c). However, the coupling \(g_{{\rm{2q}}}^{{\rm{ff}}}\) between the flipflop qubits can be kept essentially constant by simply readjusting V _{t}, using, e.g., the method described in Fig. 2f, g. Figure 5d shows that one can keep a constant value of, e.g., \(g_{{\rm{2q}}}^{{\rm{ff}}} = 1\) MHz for any interdonor spacing between 180 and 500 nm, by adjusting V _{t} between 11.3 and 11.8 GHz. In other words, since the flipflop qubit coupling is mediated by a tunable interaction with their respective charge qubits, the interqubit interaction does not need to decay with r ^{3}, as one would otherwise get when the dipole interaction couples the qubits directly^{26, 31}. Therefore, twoqubit operations can be turned on between pairs of qubits separated by many sites in a twodimensional array. This tunable longrange connectivity can be exploited to great advantage in largescale quantum processors^{50}. The large tolerance in g _{dd} also accommodates very well the donor depth uncertainties inherent to ion implantation^{45}, given the linear dependence of \(g_{{\rm{2q}}}^{{\rm{ff}}}\) on d _{i} (Eqs. (16) and (17)).
We conclude that our scheme provides a dramatic reduction in the fabrication complexity, especially compared to schemes that require placing a gate between a pair of tightly spaced donors, such as the Kane’s proposal^{15}, which requires r ≈ 15 nm separation between two ^{31}P nuclear spins. Note that, by relocating the problem of valley oscillations from the exchange interaction^{15} to the tunnel coupling, we have effectively provided a way in which the delicate parameter can now be tuned using a much simpler gate geometry.
Scaling up using circuit quantum electrodynamics
In order to reach the longterm goal of a largescale quantum processor, wiring up the control and readout lines for each individual qubit is not trivial, given the high density in typical spin qubit architectures^{51}. Recent solutions include crosswiring using multilayer lithography^{26} or floating gate electrodes inspired by dynamic random access memory systems^{52}. In both cases, using flipflop qubits with longdistance interactions would result in widely spaced donors and loose fabrication tolerances. In addition, since flipflop qubits are coupled via electric fields, they could be spaced further apart by using electrical mediators. These include floating metal gates^{53} or even microwave resonators. Indeed, the use of electric dipole transitions allows a natural integration of donorbased spin qubits into a circuitquantum electrodynamics architecture^{43, 54,55,56} (see Fig. 7c for a possible device layout).
A full quantum mechanical treatment yields a chargephoton coupling rate given by Eq. (14), with ν _{E} now representing the resonator fundamental mode frequency and E _{ac} the resonator vacuum field, E _{vac}. Again, it is best to have the chargeexcited state detuned from the flipflop transition and resonator photon (see Fig. 7b), therefore minimizing charge excitation while retaining a secondorder flipflop photon coupling given by Eq. (13). Assuming δ _{so} ≈ δ _{E} ≈ 10g _{so} ≈ 10g _{E}, a d = 15 nm deep ^{31}P flipflop qubit would be coupled to photons at a \(g_{\rm{E}}^{{\rm{ff}}} \approx 3\) MHz rate. This is three orders of magnitude faster than the electronspin coupling rate to a resonator via its magnetic vacuum field^{57, 58}, and comparable to the coupling strength obtained by using strong magnetic field gradients^{59, 60}, but without the need to integrate magnetic materials within a superconducting circuit. This assumes a vacuum field amplitude E _{vac} ≈ 30 V m^{−1}, which can be obtained by using tapered coplanar waveguide or highinductance resonators^{61}.
The possibility of coupling the qubits to microwave photons provides a path for dispersive qubit readout, as well as for photonic interconnects. Nearquantum limited amplifiers have recently become available to obtain excellent readout speed and fidelities^{62}. The resonator can also be used as a quantum bus to couple two spin qubits separated by as far as 1 cm (Fig. 7c), a distance given by the mode wavelength. Figure 7b shows the detailed energylevel diagram. To avoid losses from photon decay, the qubits should be detuned from the resonator by an amount much greater than the qubitphoton coupling rates. Assuming \(\delta _{\rm{E}}^{{\rm{ff}}} = 10g_{\rm{E}}^{{\rm{ff}}}\), where \(\delta _{\rm{E}}^{{\rm{ff}}} = {\nu _{\rm{E}}}  {\epsilon _{{\rm{ff}}}}\), the effective twoqubit coupling \(g_{{\rm{2q}}}^{{\rm{ff}}} \approx {\left( {g_{\rm{E}}^{{\rm{ff}}}} \right)^2}{\rm{/}}\delta _{\rm{E}}^{{\rm{ff}}} \approx 0.3\) MHz yields a \(\sqrt {i{\rm{SWAP}}} \) gate that takes only 0.4 μs.
Discussion
Figure 7a summarizes the key figures of merit of a quantum processor based on flipflop qubits coupled by electric dipole interactions. Fast onequbit x(y)gates are attainable with low electric drive power and error rates ~10^{−3}. Twoqubit \(\sqrt {i{\rm{SWAP}}} \) gates are fast and with error rates approaching 10^{−3}. At the end of all operations, the phase of each qubit can be corrected, via adiabatic zgates, in fast time scales and low error rates ~10^{−4}. These values are based on current experimentally known values of charge noise in silicon devices^{39}, and are possibly amenable to improvement through better control of the fabrication parameters. More advanced control pulse schemes could allow for faster gates with less leakage^{63,64,65}, and active noise cancellation techniques, e.g., pulses for gate time jitter^{66} or decoherence^{67} suppression, could further improve gate fidelities.
Idle qubits are best decoupled from all other qubits by having the electron at the interface and the quantum state stored in the nuclear spin, which has a record coherence times T _{2} ≳ 30 s^{7}, and can be even longer in bulk samples^{6}. Quantum information can be swapped between the nuclear and the flipflop qubit by simply applying an ESR πpulse that excites the \(\left { \downarrow \Downarrow } \right\rangle \) state to \(\left { \uparrow \Downarrow } \right\rangle \) (Fig. 1c).
Qubit readout can be obtained by spindependent tunneling into a cold charge reservoir, detected by a singleelectron transistor^{18}. Readout times can be ~1 μs with cryogenic amplifiers^{68}, which is comparable to the time necessary to perform, e.g., ~20 individual gates lasting ~50 ns each, in a surface code error correction protocol^{2}.
A largescale, faulttolerant architecture can be built in a variety of ways. One or twodimensional arrays can be built to implement error correction schemes such as the Steane^{69} or the surface^{2} code, since all mutual qubit couplings are tunable and gateable. A larger processor can include a hybrid of both coupling methods, incorporating cells of dipolarly coupled qubits, interconnected by microwave photonic links (Fig. 7d), in which case more advanced errorcorrection codes can be implemented^{1, 3, 4, 50}. Microwave resonators could be also used to interface donors with superconducting qubits^{8, 70}, for the longterm goal of a hybrid quantum processor that benefits from the many advantages of each individual architecture^{55}.
In conclusion, we have presented a way to encode quantum information in the electronnuclear spin states of ^{31}P donors in silicon, and to realize fast, highfidelity, electrically driven universal quantum gates. Our proposal provides a credible pathway to the construction of a largescale quantum processor, where atomicsize spin qubits are integrated with silicon nanoelectronic devices, in a platform that does not require atomicscale precision in the qubit placement. The qubits are naturally amenable to being placed on twodimensional grids and, with realistic assumptions on noise and imperfections, are predicted to achieve error rates compatible with faulttolerant quantum error correction.
Methods
Adiabaticity
Given a timedependent Hamiltonian in a twodimensional Hilbert space,
in units of rad s^{−1}, the adiabatic condition is expressed as^{71}
where \({\omega _{{\rm{eff}}}} = \sqrt {{\Delta ^2} + {\Omega ^2}} \) is the instantaneous transition angular frequency between eigenstates, and \(\dot \alpha \) is the rate of change of the orientation of ω _{eff} (α = arctan(Ω/Δ)). It follows from Eq. (20) that
Although the processes described in this paper involve multiple levels, we applied Eq. (21) in different forms as an approximation of adiabaticity. This was confirmed to be always valid by checking that the leakage errors were kept below a target level.
In particular, for onequbit zgates and twoqubit \(\sqrt {i{\rm{SWAP}}} \) gates, we used \({\Delta _{\rm{c}}} = \pi e\left( {{E_{z}}  E_{z}^0} \right)d{\rm{/}}h\) and Ω_{c} = πV _{t} to find K _{c} for the charge qubit, and Δ_{so} = πδ _{so} and Ω_{so} = 2πg _{so} to find K _{so} for the spincharge coupling. For a chosen adiabatic factor K, we find E _{z}(t) by satisfying the condition min(K _{so}, K _{c}) = K.
For onequbit drive, we used Δ_{E} = πδ _{E} and Ω_{E} = 2πg _{E} to find K _{E}. A particular choice of K = K _{E} sets the adiabatic sweep rate of E _{ac}(t).
Estimation of dephasing and gate errors
In order to estimate the effects of quasistatic E _{z} noise on dephasing, we first calculate the flipflop qubit transition frequency \({\epsilon _{{\rm{ff}}}}\) (difference between eigenfrequencies corresponding to eigenstates closest to \(\left {g \downarrow \Uparrow } \right\rangle \) and \(\left {g \uparrow \Downarrow } \right\rangle \), which we denote as \({\left {g \downarrow \Uparrow } \right\rangle _{\rm{e}}}\) and \({\left {g \uparrow \Downarrow } \right\rangle _{\rm{e}}}\)). Next, for a uniformly distributed noise in the range \(E_{z}^n = \sqrt 3 \left[ {  E_{z,{\rm{rms}}}^{{\rm{noise}}},E_{z,{\rm{rms}}}^{{\rm{noise}}}} \right]\), we estimate the qubit dephasing rate to be
where N _{ n } is the number of sampled \(E_{z}^n\) and \(\epsilon _{{\rm{ff}}}^n\) is calculated for each value of \(E_{z}^n\).
The averaged error rate (without noise) of a desired adiabatic unitary process U _{ideal} is calculated by averaging the fidelity of the actual process U over a set of initial states \(\left j \right\rangle \),
where N _{ j } is the number of initial states. For onequbit gates (e.g., a π zgate or a π/2 x(y)gate), we choose \(\left j \right\rangle = \left\{ {{{\left {g \downarrow \Uparrow } \right\rangle }_{\rm{e}}},{{\left {g \uparrow \Downarrow } \right\rangle }_{\rm{e}}},\left( {{{\left {g \downarrow \Uparrow } \right\rangle }_{\rm{e}}} + {{\left {g \uparrow \Downarrow } \right\rangle }_{\rm{e}}}} \right){\rm{/}}\sqrt 2 ,\left( {{{\left {g \downarrow \Uparrow } \right\rangle }_{\rm{e}}} + i{{\left {g \uparrow \Downarrow } \right\rangle }_{\rm{e}}}} \right){\rm{/}}\sqrt 2 } \right\}\) and N _{ j } = 4, whereas for twoqubit gates (e.g., \(\sqrt {i{\rm{SWAP}}} \)) \(\left j \right\rangle = \left {{j_1}} \right\rangle \otimes \left {{j_2}} \right\rangle \) (the 1,2 indexes refer to the aforementioned four initial states for each qubit) and N _{ j } = 16.
To estimate the averaged gate error rate under quasistatic E _{z} noise, the actual process U and eigenstates \(\left j \right\rangle \) are calculated for each value of \(E_{\rm{z}}^n\) before averaging,
Finally, to estimate errors due to chargephonon relaxation, we multiply the averaged charge excitation by its relaxation rate and assume a exponential decay in fidelity:
where \(\left j (t)\right\rangle\) are the timeevolution of the initial set states \(\left j \right\rangle\). For twoqubit gates, we sum up the error rate of each qubit.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
We thank A. Blais, H. Bluhm, M. Eriksson, J. O’Gorman, S. Benjamin, A. Pályi, J. Salfi, M. Veldhorst, A. Laucht, R. Kalra, and C.A. ParraMurillo for discussions. This research was funded by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027), the US Army Research Office (W911NF1310024), and the Commonwealth Bank of Australia. Tightbiding simulations used NCN/nanohub.org computational resources funded by the US National Science Foundation under contract number EEC1227110.
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A.M. and G.T. conceived the project. G.T. developed the theoretical framework, with F.A.M.’s assistance and under A.M.’s supervision. G.T. and F.A.M. performed calculations and numerical simulations with S.T.’s and V.S.’s assistance. R.R. and G.K. developed the qubit simulation capabilities in the NEMO3D code. G.T., A.M., and F.A.M. wrote the manuscript, with input from all coauthors.
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Tosi, G., Mohiyaddin, F.A., Schmitt, V. et al. Silicon quantum processor with robust longdistance qubit couplings. Nat Commun 8, 450 (2017). https://doi.org/10.1038/s4146701700378x
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