Fast Noise-Resistant Control of Donor Nuclear Spin Qubits in Silicon

Fast noise-resistant control of donor nuclear spin qubits in silicon

James Simon1,2, F. A. Calderon-Vargas2, Edwin Barnes2, and Sophia E. Economou2∗ 1Department of Physics, University of California, Berkeley, California 94720, USA 2Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

A high degree of controllability and long coherence time make the nuclear spin of a phosphorus donor in isotopically puriﬁed silicon a promising candidate for a quantum bit. However, long- distance two-qubit coupling and fast, robust gates remain outstanding challenges for these systems. Here, following recent proposals for long-distance coupling via dipole-dipole interactions, we present a simple method to implement fast, high-ﬁdelity arbitrary single- and two-qubit gates in the absence of charge noise. Moreover, we provide a method to make the single-qubit gates robust to moderate levels of charge noise to well within an error bound of 10−3.

I. INTRODUCTION on clock transitions. Our approach is based on using optimally designed control pulse waveforms and energy Nuclear spins in the solid state present unparalleled transition modulation. Accordingly, we derive a time- advantages as a platform for quantum computing due independent analytical approximation for the system’s to their long coherence times [1, 2] and high degree of Hamiltonian, explain how to rapidly implement arbi- controllability [3–5]. In particular, the nuclear spin of a trary, noise-resistant single-qubit gates with fidelities ex- phosphorus donor in silicon is a promising candidate for a ceeding 99.9% even in the presence of significant charge quantum bit [6, 7] owing to its coherent control [8–10] and noise, and provide a method to use the dipole-dipole in- minute-long coherence time [2]. The use of isotopically- teraction to implement cphase gates across a distance purified silicon nanostructures [11] considerably reduces of 0.5 µm. The single-qubit and cphase gates have max- magnetic environmental noise, allowing high-fidelity con- imum durations of 500 ns and 750 ns, respectively, with trol [8]. However, long coherence times are only use- special cases such as single-qubit Z gates being much ful when gates are very fast in comparison. One of faster (< 25 ns). An advantage of our protocol compared the difficulties of using the nuclear spin as a qubit has to prior work [16] is that it does not require finely tuning been the implementation of fast single- and two-qubit the system to a clock transition. This is not done at the gates. Controlling a nuclear spin with an oscillating expense of gate performance, and our gates are as robust magnetic field as in nuclear magnetic resonance is slow, but faster than those of Ref. 16. with typical gate times ranging from a few to tens of mi- The paper is organized as follows. In Sec. II we in- croseconds [8, 9, 12]. Moreover, most of the approaches troduce the system and its Hamiltonian. In Sec. III, we for multi-qubit operations require short interaction dis- derive an analytical time-independent Hamiltonian fol- tances [6][13], and thus demand near-atomic precision in lowing two approaches. In Sec. IV, we define the qubit the placement of the donors [14, 15]. states and explain how to implement robust single-qubit To overcome these challenges, Ref. 16 proposes the dy- gates, specifically arbitrary Z-rotations and X-rotations. namical creation of a strong electric dipole transition at We give a method for implementing fast controlled-phase microwave frequencies for the nuclear spin by sharing an gates between two adjacent qubits separated by a dis- electron between the donor and the Si/SiO2 interface and tance of 0.5 µm in Sec. V. We conclude in Sec. VI. applying an oscillating magnetic field. This facilitates the implementation of two-qubit gates via dipole-dipole interactions or, alternatively, the qubit’s coupling to other II. THE SYSTEM quantum systems. Moreover, nuclear spin-flip transitions can also be sped up by including an oscillating electric arXiv:2001.10029v2 [quant-ph] 20 May 2020 field along with the magnetic drive. A potential down- The system follows the experimental proposal reported in Refs. 16 and 17, where a donor 31P atom is embed- side of making the system amenable to electrical control 28 is that it increases its sensitivity to charge noise due to ded in enriched Si a distance d away from a Si/SiO2 the charge component in the states encoding the qubit. interface, as shown in Figure 1. The donor atom pro- vides a nuclear spin I = 1/2 with gyromagnetic ratio However, Ref. 16 shows that there are regions in param- −1 γn/2π = 17.23 MHzT and a free electron with spin eter space (“clock transitions”) where the nuclear spin −1 transition is insensitive to electrical noise to at least first S = 1/2 and gyromagnetic ratio γe/2π = 27.97 GHzT . order. The electron and nuclear spins are coupled via a hyper- In this work, we propose an alternative path toward fine interaction with coupling strength A, which is ap- robust high-fidelity single-qubit gates that does not rely proximately equal to 117 MHz when the electron is bound to the nucleus. A metal gate positioned on top of the donor atom is used to control the position of the electron via electric fields, which also tunes the hyperfine inter- ∗ [email protected] action A down to zero when the electron is at the inter- 2

Note also that all-electrical spin control is possible even in the presence of a signiﬁcant spin-orbit interaction, but it is vulnerable to charge noise [20]. The Hamiltonian of the system, therefore, consists of the orbital part, the Zeeman part, and the hyperﬁne coupling:

de (∆E + Ea cos [ωEt]) id Vt id Horb = − τz + τx , 2~ 2 1 id − τz HB =B0 γe 1 + ∆γ Sz − γnIz 2 (2) 31 FIG. 1. One nuclear spin qubit in the system described. A P + B cos [ω t](γ S − γ I ) , donor is embedded in 28Si, and the free electron in the system a B e x n x can be pulled towards a Si-SiO interface. The electron orbit 1 − τ id 2 H =A z S · I, is quantized into a |di state on the donor and a |ii state on the A 2 interface, and both the electron and nuclear spins are used. Static and oscillating electric and magnetic fields are used where ∆E = E − E0 is the deviation of the electric field to control the system. The qubit is ultimately stored in the away from the ionization point. nuclear spin state, with the other degrees of freedom used for The qubit is encoded in the two lowest-energy eigen- driving gates. states of the system, which, in the absence of AC driving, are approximately |g ↓⇑i and |g ↓⇓i, where |gi is the ground eigenstate of the orbital part of the Hamiltonian face. Moreover, the gyromagnetic ratio of the electron with no AC fields. Therefore, it is convenient to express bound to the nucleus can differ from that of an electron the total Hamiltonian in (1) in a basis spanned by the at the interface by an amount ∆γ that can reach up to orbital eigenstates {|gi , |ei}. The electron position oper- id id 0.7% [18]. Therefore, the Hilbert space includes three ators τx and τz in the orbital eigenbasis are binary degrees of freedom: the nuclear spin of the donor id de∆E Vt atom, the spin of the free electron, and the position of τ = τz + τx, z ε ε the free electron, which is quantized into a state on the ~ 0 0 (3) donor atom, |di, and a state at the interface, |ii, which id Vt de∆E τx = − τz + τx, is a good approximation as demonstrated by Ref. 17. ε0 ~ε0

The system has two control fields, one electric and one where τz = |gi hg| − |ei he| and τx = |gi he| + |ei hg| are magnetic, each with static (DC) and oscillating (AC) p 2 2 the orbital operators, and ε0 = Vt + (de∆E/~) is the components. The DC component of the electric field, orbital (charge) energy splitting. The Hamiltonian com- E, points along the donor-interface axis and controls ponents in Eq. 2 have the following form in the basis the electron position (see Fig. 1), determining the am- spanned by the orbital and spin eigenbasis: plitudes of the states |ii and |di in the orbital ground state. The AC electric field is parallel to E and is given −ε0 deEa cos(ωEt) de∆E Vt Horb = τz − τz + τx , by Eac(t) = Ea cos(ωEt). In particular, when E = E0, 2 2~ ~ε0 ε0 where E is the magnitude of the electric field at the ion- 1 0 1 de∆E/~τz + Vtτx ization point, the orbital ground state of the electron has HB =B0γe + − ∆γ Sz 2 2ε0 equal probability to be at the donor nucleus and at the − B0γnIz + Ba cos [ωBt](γeSx − γnIx) , interface. When E E0, the electron is fully on the 1 donor (|di), and when E E0 the electron is pulled off de∆E Vt HA =A − τz − τx S · I. the donor (|ii). The transition frequency between the or- 2 2~ε0 2ε0 bital ground and excited states at the ionization point is (4) equal to the tunnel coupling Vt. A strong static magnetic field B0 (B0(γe + γn) A) splits the energy of the nuclear and electron spin states ({|⇑i , |⇓i} and {|↑i , |↓i}, III. DERIVING THE TIME-INDEPENDENT respectively). The AC magnetic field is perpendicular to HAMILTONIAN B0 and is given by Bac = Ba cos(ωBt). The static electric and magnetic fields are parallel to avoid reductions The dominant energy scales of this system are the in spin relaxation times caused by spin orbit effects [19]. charge splitting ε0 and the electron spin splitting B0γe, 3 which are driven at frequencies ωE and ωB, respec- down, so these states are approximately equal to |g ↓⇑i tively. We transfom into a rotating frame involv- and |g ↓⇓i. We refer to the exact qubit states as |⇑i˜ ing both frequencies, leaving a Hamiltonian that is and |⇓i˜ , respectively. With zero AC fields and typi- largely static. Therefore, we move the Hamiltonian cal parameter values (given below), the approximations ˜ † ˙ † to the rotating frame H = ΛHΛ − iΛΛ with Λ = |⇑i˜ ≈ |g ↓⇑i and |⇓i˜ ≈ |g ↓⇓i hold to within an overlap exp [−it (ωE(τz/2 + Iz) − ωB(Sz + Iz))], where the sys- error of at most 10−4, which becomes much lower still Eade tem’s dominant off-diagonal energy terms (Baγe, ~ when ∆E > 0. and AVt ) become effectively static. Assuming that the ε0 The dominant source of decoherence in this system driving fields’ detunings are small and the orbital and is quasi-static charge noise with a typical 1/f spec- electron spin splittings are similar (i.e. ε0 ≈ ωE ≈ trum [18, 22]. We model the quasi-static charge noise as B0γe ≈ ωB), we can then apply the rotating wave acting along the z-axis, directly perturbing the applied approximation (RWA), dropping all rapidly oscillating electric field ∆E [23]. The energy splitting δq between terms from H˜ to get: the two qubit states depends on the applied electric field (see Figure 2), so noise in the electric field will lead to −ε0 + ωE deVt uncertainty in the energy difference of the qubit states, H˜0 = τz − Ea(t) τx + (B0γe − ωB)Sz 2 4~ε0 causing dephasing. This energy splitting can be very ac- 1 de∆Eτz curately approximated by − (B0γn + ωB − ωE)Iz + B0γe∆γ − Sz 2 2~ε0 hAi δq ≡ E˜ − E˜ ≈ B0γn + , (6) Ba(t)γe A de∆E ⇓ ⇑ 2 + Sx + 1 − τz SzIz 2 2 ~ε0 where E⇓˜ and E⇑˜ are the energies of the respective states AVt 2 − (|g ↑⇓i he ↓⇑| + |e ↓⇑i hg ↑⇓|) . and hAi = A|hg|di| = (A/2)(1−de∆E/ε0). If the system 4ε0 idles for a time t with a small error in the electric field (5) δE, we expect the dephasing to be approximately given Except for the final term coupling the |g ↑⇓i and |e ↓⇑i by R − dδq δEt . states, this Hamiltonian is entirely diagonal when the z d∆E driving fields are zero. Following Ref. 16, in all the following calculations we −1 This approximate RWA Hamiltonian works very well use A/2π = 117 MHz, γe/2π = 27.97 GHz T , γn/2π = for short times. However, as the system evolves, the ap- 17.23 MHz T−1, and ∆γ = −.002, and choose d = 15 nm, proximate evolution gradually becomes dephased relative B0 = 0.2 T, and Vt = B0(γe + γn). Moreover, we choose to the true evolution due to energy shifts caused by the ∆E = 104 V m−1 so that in the absence of AC driving, dropped high-frequency terms, in a similar effect to the the qubit idles with the electron at the interface, in a Bloch-Siegert shift [21]. Corrections to the RWA are re- region where dδq/d∆E is very small and dephasing is thus quired, and thus we use multi-frequency Floquet theory reduced (see Appendix B for further discussion). Every to construct a Floquet Hamiltonian HF that takes into gate shown here starts and ends at this chosen idling account these higher-frequency modes. Then we use 2nd- point. We also define our gates in the idling frame of the order quasi-degenerate perturbation theory to reduce it qubit, so the evolution while idling is simply the identity. to a (still non-oscillating) approximation H0 that repro- Any single-qubit gate can be decomposed into the form duces the evolution given by the oscillating lab-frame Rz(θz1)Rx(θx)Rz(θz2), where a Z rotation is defined as θ Hamiltonian with typical fidelity > 0.9999 for the gates Rz(θ) = exp −i 2 σz , θ ∈ [0, 2π), and an X rotation is θ we describe in this paper. The derivation of this approx- defined as Rx(θ) = exp −i 2 σx , θ ∈ [0, π). The qubit imation is detailed in Appendix A. Pauli operators are defined following σ = |⇑ih˜ ⇑|−|˜ ⇓ih˜ ⇓|˜ . ˜ z As with H0, this approximation’s only time depen- Below we show how to implement a noise-resistant Rz(θ) dence is in the changing envelopes of the control pulses, gate. making it both conceptually simpler and faster to sim- ulate with common software. It is thus useful for opti- mizing gates or running accurate simulations much faster A. Rz(θ) Gates than with the lab-frame Hamiltonian. In this paper, we use the lab-frame Hamiltonian in all of our final results Figure 2 shows that simply changing ∆E changes δq, for the sake of caution, but we have confirmed that using causing a phase difference between the qubit states to ac- this approximation gives the same results. cumulate. If leakage were not a concern, one could implement an effectively noiseless Z rotation by shifting from the idling condition of ∆E 0 to ∆E 0. δq then shifts IV. QUBIT STATES AND GATES by ∼ 2π · 60 MHz, causing a full 2π rotation in ∼ 20 ns, and dδq/d∆E will be small as during idling. However, The qubit is defined to be the two lowest-energy eigen- the non-oscillating electric field ∆E cannot be shifted ar- states of the system. In the absence of driving, the elec- bitrarily fast, and changing ∆E changes the transforma- tron is predominantly in the ground orbital with spin tion we have used in going from the {|ii , |di} basis to the 4

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FIG. 2. Energy splitting δq between qubit states in the lab FIG. 3. Examples of ∆E pulses used to implement Rz(θ) frame as a function of ∆E with no driving ﬁelds. Both the gates for various values of the total gate time. For long times, numerical value and the approximation given in Eq. 6 are ∆E is moved to its minimum value, is held for several ns, shown. Varying ∆E changes this splitting, allowing the im- and returns. For short times, ∆E cannot be changed fast plementation of Rz(θ) gates. enough to reach the minimum while remaining adiabatic, so a shallower pulse is used.

{|gi , |ei} basis, causing another term to appear in H (as * derived in Appendix C) that can drive the system out of the logical space. Nonetheless, the effect on the fidelity - /- is negligible for shift times of a few ns, so this does not - θ significantly affect the gate time. The resulting evolution -. /- !" will be adiabatic in the qubit subspace, so we only need #$$%&'( ") to consider the phase accumulated. -- We can calculate the phase accumulated using * + ,* ,+ -* -+ Z T / (01) 0 θ = − (δq(t) − δq )dt, (7) 0 FIG. 4. Exact and approximate Rz(θ) angles for different 0 gate durations T using the gate scheme described in Section where δq is the qubit splitting while idling, which we sub- IV A. The approximate curve is found from the approxima- tract so we work in a frame where, at idling, the evolution tion in Eq. 6, integrated according to Eq. 7. The exact and operator is the identity. For the pulse shapes, we use a approximate curves agree to within an angle of .08 rad. cosine window function w defined as follows:  [1 − cos(πt/τ)]/2 0 ≤ t < τ,  is generated in T ≈ 14 ns. The numerically calculated 1 τ ≤ t < T − τ, w(t, τ, T ) = phase is quite close to a simple approximation obtained [1 − cos(π(T − t)/τ)]/2 T − τ ≤ t ≤ T,  using Eqs. 6 and 7, demonstrating that the phase of the 0 t < 0 or t > T. gate is accumulating as we describe. While these gates (8) are quite fast, we note that the 5 ns ramping time of our When implementing an Rz(θ) gate as described above, pulses is well within the risetime limitations of typical the varying function ∆E(t) is constrained by the fact that waveform generators. In the event that timing errors ∆E has to change slowly enough for evolution to be adi- become an issue, these Rz(θ) gates could be combined abatic. We choose a minimum time of 5 ns to move from with the Rx(θ) gates described below to construct a BB1 idling to minimum ∆E, so, for Rz(θ) gates with total sequence [24] to correct over-rotation errors. time T shorter than 10 ns, there will not be time for ∆E As mentioned before, charge noise is the most dele- to reach its minimum value, so we simply make the ∆E(t) terious source of error for this type of system [17, 25]. pulse shallower. We choose ∆E(t) = ∆Eidle−S·w(t, τ, T ), Although this noise has been measured to have a 1/f 4 −1 where Eidle = 10 V m , τ = min(5 ns,T/2) and S = power spectrum [18], this noise is largely concentrated (2 × 104 V m−1) · min(1,T/(10 ns)). Figure 3 shows sev- at frequencies below 1 kHz. Because our gate times are eral examples of ∆E(t) for various T . As no AC fields several orders of magnitude faster than the noise fluctua- are needed, we choose Ea(t) = Ba(t) = 0. tion timescale, it should be a good approximation to treat The angle of the implemented Rz(θ) gate as a function the noise as quasi-static [26]. In this work, we model this of T is plotted in Fig. 4. An arbitrary Z-rotation can noise by adding a constant stochastic error δE to ∆E for be generated in under 25 ns, and a Z-gate with θ = π the duration of a gate. Since the charge noise is statis- 5

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FIG. 5. Gate infidelity vs noise strength for three Rz(θ) gates with θ = −π/4 (T = 6.632 ns), θ = π (T = 13.560 ns), and θ = 2π (T = 22.116 ns). The numerical error in our simula- FIG. 6. A level diagram showing the states of the system in tions is roughly 10−6. the original, lab basis and the significant off-diagonal terms (both oscillating and static). While idling, the qubit states are the lowest two states. There are other off-diagonal terms involving A and Ba, but they are far off resonance and are of tical in nature, we draw δE from a normal distribution minor significance, contributing to correction terms in H0. with standard deviation σδE and report average infidelity over the distribution. The gate infidelity is defined as [27] 1 † † 2 drive a transition adiabatically with respect to the qubit 1 − F = 1 − n(n+1) [Tr U U + | Tr U0 U | ], where n is the Hilbert space dimension, U is the generated gate, and subspace. If ∆E and B0 are chosen so that the lab- frame energies of |g ↑⇓i and |e ↓⇑i are similar (i.e., U0 is the desired gate. Figure 5 shows the gate infidelity for three different angles of rotation. For typical noise ε0 ≈ B0(γe + γn)), the hyperfine coupling strongly hy- with an r.m.s. of 100 V m−1 [17], the error is well below bridizes these states. Transitions between the qubit −4 states can then be seen as similar to Raman transitions 10 for all angles. Our Rz(θ) scheme is quite noise- resistant as it is, because the system spends most of its [28], but with two intermediate states instead of one. The driving field amplitudes and frequencies can be set time with ∆E 0 or ∆E 0, so that dδq/d∆E is small for most of the gate. like in an adiabatic Raman transition (i.e. making the driving matrix elements equal and the large detunings from intermediate states the same) to give a transition close to an X-gate with a leakage probability below 10−4. B. Rx(θ) Gates The large detunings separate the qubit subspace from the rest of the Hilbert space, and we use slowly-varying An R (θ) gate requires a coupling between the two x pulse shapes instead of square pulses, so the resulting qubit states. The simplest way to achieve this is by driv- transition is very adiabatic. The driving amplitudes and ing B on resonance with the nuclear spin, but the scale ac frequencies can then be optimized while worrying only of γ makes this slow, with typical times ranging from n about the computational subspace, while also using the a few to tens of microseconds for π rotations [8, 9, 12]. R (θ) gates described previously to cancel unwanted ac- A faster solution in this system is to create an effective z cumulated phases, giving a high-fidelity R (θ) for any coupling through intermediate non-computational states. x given angle. Figure 6 shows a level diagram of the system including This gate scheme, however, is extremely sensitive to all strong, near-resonance couplings between states, as electrical noise. This can be seen by writing out the reflected in H˜ . The simplest way to drive fast transi- 0 three relevant transition energies as follows: tions between the qubit states is to use two intermediate states: δ⇓ ≡ E|g↑⇓i − E|g↓⇓i ≈ B0γe − hAi/2,

Bac A Eac δ ≡ E − E ≈ ε − A/4 + hAi/2, |g ↓⇓i ⇐==⇒ |g ↑⇓i ⇐==⇒ |e ↓⇑i ⇐==⇒ |g ↓⇑i. ⇑ |e↓⇑i |g↓⇑i 0 δmid ≡ E|e↓⇑i − E|g↑⇓i ≈ ε0 − B0(γe + γn) − A/4 + hAi/2. The first major problem one encounters is the fact (9) that, if the driving frequencies are simply chosen to All three of these transition frequencies depend on ∆E. be on resonance, there will be severe leakage into the In addition, ε0 strongly depends on ∆E when |∆E| 0, non-computational states. The complexity of the sys- while hAi strongly depends on ∆E when ∆E ≈ 0, so there tem and the limited number of controllable parameters is no value for ∆E that ameliorates this problem. If any would seem to make this a difficult problem. However, of the three transitions are far off resonance, the effective with slightly different choices of driving fields, one can coupling between the qubit states approaches zero and 6

and the three angles of this decomposition vary continu- ously with U except when θx = 0, π due to a phenomenon called gimbal lock. Except at these points, then, to first order, an arbitrary gate’s dependence on charge noise can 0 0 be decomposed as Rz(θz1 +θz1δE)Rx(θx +θxδE)Rz(θz2 + 0 0 θz2δE). The effect of this sweep will be to make θx ≈ 0, eliminating the noise-dependence of the Rx component. FIG. 7. The electric field ∆E during a sweep gate, with and Additionally, tweaking the start and end points of the 0 0 without static error δE in the electric field. The resulting sweep can change θz1 and θz2. In the special case of 0 0 evolution operators are shown. As explained in the text, a θx = π, θz1 = θz2, the two error terms cancel because key fact is that the middle stretch of the evolution, U(−D + Rz(θz)Rx(π) = Rx(π)Rz(−θz), and the noise depen- δE, D), is the same with and without noise, leaving only small dence is eliminated. Gimbal lock is not a problem here differences on the edges that constitute Rz errors. 0 0 because in practice θx 6= π, so θz1 and θz2 do not diverge, and even at θx = π they diverge together and the divergences cancel. This means that to create an X-gate, an X-rotation becomes impossible. Thus, while the two we can apply a Rz(−θz1 + θz2) gate after the sweep to qubit states are effectively coupled, ∆E has to be known remove the residual Rz gates and leave only Rx(π) ≡ X. to high precision. We use numerical simulations in conjunction with the Our solution to this is to have ∆E sweep through a Euler decomposition to determine the driving parame- broad range of values at a fixed rate instead of remaining ters necessary to produce this cancellation. constant. The probability transfer between the two qubit Our precise control protocol for implementing a noise- resistant X-gate is summarized as follows. We sweep states will happen in a short period in the middle when- −1 −1 ever ∆E ≈ 0. If there is quasi-static charge noise, the ∆E from −2000 V m to 2000 V m in 110 ns using the constant sweep rate ensures that this transition will still function l: happen identically, but it will simply be shifted slightly y t/τ 0 ≤ t < τ , in time, and there will be only noisy Z-rotations before  1 1 1  t−τ1 y1 + (y2 − y1) τ1 ≤ t < τ2, and after the gate. To explain this, consider sweeping ∆E τ2−τ1 l(t, τ1, y1, τ2, y2,T ) = T −t from −D to D at a constant rate, first without error in y2 τ2 ≤ t ≤ T,  T −τ2 ∆E, and second with an error of δE > 0, so ∆E actually 0 otherwise. sweeps from −D + δE to D + δE. In both cases, there (10) is a segment of the evolution in which ∆E sweeps from Then the control pulse is ∆E(t) = ∆Eidle + −D + δE to D. The difference is that in the first case, l(t, τ1, −∆Eidle −D, τ1 +τs, −∆Eidle +D, 2τ1 +τs), where −1 there is a sweep from −D to −D+δE before this segment, ∆Eidle = 10 000 V m is the idling voltage, D = −1 and in the second case, there is a sweep from D to D+δE 2000 V m is the amplitude of the sweep, τ1 = 5 ns is the after it. Far from ∆E ≈ 0, the evolution operator will be setup time and τs = 110 ns is the duration of the sweep. diagonal in the qubit subspace, so these small pieces are For the AC driving fields, we use the window functions −1 2 just Rz rotations, so the effect of this error δE is just from Eq. 8 and choose Ea(t) = λ · (255.2 V m ) · w (t − 2 to introduce phase errors before and after the gate. To τ1, τ2/5, τ2), Ba(t) = λ · (33.26 mT) · w (t − τ1, τ2/5, τ2), explain why, consider a gate that sweeps from ∆E = −D where τ2 = τ1 + τs and ωE = ε0 − 2π · 232.428 MHz, to ∆E = D at a constant rate. Let U(a, b) represent the ωB = B0γe − A/4 − 2π · 217.096 MHz. Squaring the win- evolution operator resulting from sweeping ∆E through dow function produces pulses that turn on more gradu- the range [a, b]. As shown in Fig. 7, in the ideal case, we ally, improving adiabaticity. We include the free param- get the evolution U0 = U(−D + δE, D)U(−D, −D + δE), eter λ to control θx; the amplitudes have been chosen so while if there is a static charge noise δE the actual evo- that λ = 1 gives Rx(π), and λ = 0 must give θx = 0 as lution operator is UδE = U(D,D + δE)U(−D + δE, D). the driving fields are off, so varying λ ∈ (0, 1) necessarily The middle stretches of these evolution operators, from gives intermediate values of θx. All other parameters are ∆E = −D + δE to ∆E = D, are essentially identical; set to the values quoted at the beginning of Sec. IV. Plots apart from the ramping up of the AC fields, all param- of ∆E and the AC fields are given in Fig. 8. eters as a function of time are the same. If D is large, Figure 9(a) shows the effect of quasi-static charge noise then the AC electric driving is very far off resonance at on both a naive X-gate as described earlier and on our the ends of the evolution span, and there is no effective noise-resistant X-gate. Both gates take roughly 140 ns, coupling between the computational states, so the error including the corrective Z-rotations at the beginning and † operators U(D,D + δE) and U (−D, −D + δE) are en- end of the gate. The noise-resistant gate shows substan- tirely diagonal and amount to Rz errors. The effect of tially better performance, with an error well below 10−3 small charge noise, then, is only to add dephasing before for a noise with r.m.s. of 100 V m−1. and after the sweep gate. As our protocol uses adiabaticity and involves a sweep, As discussed above, an arbitrary gate U can be decom- it might seem suggestive of adiabatic passage protocols posed into Euler angle form U = Rz(θz1)Rx(θx)Rz(θz2), in NMR [29], but it is in fact quite different: here, the 7

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#! ! -!

- ! ! ! ! %! &! '! #!! #! - () - - FIG. 8. Control pulses for ∆E, Ea, and Ba during a sweep $ gate giving a Rx(π) rotation. The AC ﬁelds only turn on dur- -# (() !" ing the middle 110 ns when ∆E is steadily sweeping through #$""% zero. An R (θ) gate (not shown) is also applied before or -" #$""% & '()* z after to cancel extra phases accumulated during the sweep. -! -

- - sweep does not cause the transition, serving only to sup- - press the eﬀects of noise. The gate is also equally adia- $ (,) !" batic without it. The reason that even the naive X-gate -# #$""% scheme could have such high ﬁdelity without noise cor- -" #$""% & '()* rection is that the large detunings and slowly-changing -!

pulse envelopes ensure that the evolution is very nearly - adiabatic. Using our corrected rotating-frame Hamilto- - nian H0 (see Appendix A), we can find the eigenstates in - the middle of a gate’s evolution and examine the purity $ # " of the evolution operator in the computational subspace σδ (/) to quantify the adiabaticity. Doing so, we find that the Rx(θ) gates described above have leakage due to nonadi- −4 FIG. 9. Infidelity of simulated Rx(θ) gates for (a) θ = π, abaticity near or below 10 at any given point through- (b) θ = 3π/4, (c) θ = π/2, (d) θ = π/4. The infidelity of the out the evolution. However, one complication with our different control schemes described in the main text is plotted scheme is the fact that if ε0 ≈ 2ωE at some point while for each angle against the charge noise strength included in the oscillating electric field is on, there is a weak, sharp the simulation. We show only naive and sweep gates for θ = π two-photon resonance that excites the |gi states to |ei because the sweep Rx(π) gate is designed so that the errors states. The result of this is that, with our scheme, if ∆E due to charge noise cancel without the need for echoes. is swept over too broad a range, there will be nonadiabaticity of order 10−2 and a discrepancy between the exact and approximate evolution. With our parameters, ror. We find that the first-order approximation for the this occurs near ∆E = ±2500 V m−1, so this limits the noise-dependence of the R (θ) components of the decom- sweeping range of ∆E. z position is quite accurate (i.e. there is no need to include 00 2 We now consider the problem of a general, noise- a θz1δE /2 term, for example), indicating that correct- resistant Rx(θ) gate. The challenge is to take our noise- ing noise to first-order should give a good noise-resistant vulnerable sweep gate, which gives an evolution of the gate. form R (θ + θ0 δE)R (θ )R (θ + θ0 δE), and find a z z1 z1 x x z z2 z2 The key ingredient in our approach will be a modi- way to cancel the first-order error terms θ0 δE. Once zi fied, noise-vulnerable R (θ) gate, whose dependence on the errors are cancelled, the residual θ terms can be re- z zi charge noise we use to cancel the noise in the sweep gate. moved by R (θ) gates as before, leaving only the R (θ ). z x x If, in an R (θ) gate, we idle at a nominal value of ∆E = 0 For a given θ , we can run simulations with different z x instead of ∆E 0 (a noise-vulnerable point instead of a values of δE and decompose the resulting gates to nu- noise-resistant point according to Fig. 2), we get a rota- merically find {θ , θ0 , θ , θ0 }. To apply this proto- z1 z1 z2 z2 Adet tion error Rz δE in addition to the normal Z rota- col in specific experiments, we can use numerical sim- 4~Vt ulations to approximate these parameters, with further tion. For idling times on the order of tens of ns, this pro- optimization using the physical system to account for er- duces an error that is comparable to the dephasing errors 8

0 θzi. If these dephasing angles had the opposite sign as the First, we must derive the dipole-dipole interaction 0 θzi, the solution would be simple: one could apply a gate term. The electric dipole operator of a qubit depends of this form before and after the sweep gate with the de- only on the electron orbital. We can write the dipole 0 phasing angles equal to θzi to cancel the noise. However, operator of qubit k as the two types of error have the same sign. Our solution is to use the noise-resistant Rx(π) we constructed earlier pk = pi |ii hi| + pd |di hd| , (12) to flip the sign of this dephasing to use it to cancel the dephasing in the sweep gate. Here, we are again using where pi and pd are the effective dipoles when the elec- Rz(θz)Rx(π) = Rx(π)Rz(−θz). tron is in the |ii and |di states, respectively. Because the The full noise-resistant gate is generated as follows: electron is at the interface in the |ii state and generally near the donor nucleus in the |di state, we expect pi ≈ de corrective Rz echo Rz and pd ≈ 0. We use these approximations for the remain- z }| { z }| 0 { der of this section for simplicity. The interaction energy Rx(θx) = Rz(θz1 − ν1) Rz(ν1 + θz1δE) X 0 0 of two qubits, 1 and 2, separated by a displacement r is Rz(θz1 + θz1δE)Rx(θx)Rz(θz2 + θz2δE) (11) given by 0 XRz(ν2 + θz2δE) Rz(θz2 − ν2) . 2 | {z } | {z } p1 · p2 − 3(p1 · r)(p2 · r)/r echo Rz corrective Rz Vdip = 3 , (13) 4π0rr In creating this gate, first one chooses θx, which fixes λ and the second line of Eq. 11, which is a sweep gate. where 0 is the vacuum permittivity and r is the di- electric constant of the material ( = 11.7 for silicon). Next, the echo Rz gates are adjusted to cancel the noise r terms. All remaining phases are then cancelled by the We assume that all qubits will be fabricated with their dipoles perpendicular to the surface on which they are ar- corrective Rz gates. In practice, one can avoid the cor- rayed. In this case p ·r = 0, so the interaction simplifies rected Rz gates contained in the X-gates and simply ab- k to sorb them into the main corrective Rz gates. The major- ity of the gate duration comes from the three sweeping e2d2 |i i i hi i | gates, each contributing 120 ns to the total gate time of 1 2 1 2 Vdip = 3 . (14) 4π0rr ∼ 450 ns. The noise resistance of Rx-gates of various angles is plotted in Fig. 9. For charge noise with a r.m.s. We can then model the system with the Hamiltonian of 100 V m−1, the full sweep & echo gates all have error H = H ⊗ 1 + 1 ⊗ H + V . The lowest four eigenstates near 10−3, well over an order of magnitude better than 2q dip will be very similar to the tensor product of the indi- the naive, non-sweeping gates. vidual qubit eigenstates without Vdip, so we call them Fig. 9 shows some curves leveling out to somewhat ˜˜ ˜˜ ˜˜ ˜˜ higher infidelity at zero noise. The reason is that the {|⇑⇑i, |⇑⇓i, |⇓⇑i, |⇓⇓i} and use them as the computa- sweep gate was optimized for the extremal value of θ = tional states. Our goal is to use the dipole-dipole interac- π, with all other angles reached by interpolation. Our tion to implement a cphase gate between two adjacent optimization only reached θ ≈ 3.13 (slightly higher for qubits. We use a method similar to our Rz(θ) scheme: we the naive gate than the sweep gate), so the θ = π gates use electric fields to cause an adiabatic evolution in which have higher infidelity at zero noise. The sweep & echo the four computational states shift in energy and accumulate phases without transitions between eigenstates. gates use Rx(π) sweep gates in their pulse sequences, so they also have higher infidelity at zero noise. At realistic The evolution operator U at the end of the evolution will noise levels, this small infidelity is negligible. be of the form U =eiα|⇑˜⇑ih˜ ⇑˜⇑|˜ + eiβ|⇑˜⇓ih˜ ⇑˜⇓|˜ (15) iγ ˜˜ ˜˜ iδ ˜˜ ˜˜ V. TWO-QUBIT GATES + e |⇓⇑ih⇓⇑| + e |⇓⇓ih⇓⇓|.

If we apply the local rotations R (γ − α) to qubit 1 and The spin-charge hybridization obtained by the dis- z R (β − α) to qubit 2, and ignore a global phase of (β + placement of the electron from the donor towards the z γ)/2, we get interface induces an electric dipole that can be used for long-range coupling between qubits via a dipole-dipole U 0 =|⇑˜⇑ih˜ ⇑˜⇑|˜ + |⇑˜⇓ih˜ ⇑˜⇓|˜ interaction. As with single-qubit gates, however, it is not (16) obvious how to implement such a gate without leakage + |⇓˜⇑ih˜ ⇓˜⇑|˜ + ei(α−β−γ+δ)|⇓˜⇓ih˜ ⇓˜⇓|˜ . into the large two-qubit leakage space. Ref. 16 presents a method that uses an AC magnetic field to couple the nu- Thus if φ ≡ α −β −γ + δ is nonzero, we obtain a cphase clear spin qubit to the charge qubit, leading to an iswap gate with angle φ. gate. Here we show an alternative method, implement- Assuming that the evolution will be adiabatic, we can ing a cphase gate between two qubits with only an AC find the angles α, β, γ, δ and thus calculate φ using an electric field. expression analogous to Eq. 7, where we integrate each 9 state’s energy over the course of the evolution. Using our approximation, the energies are given by -π/ 2 2 -π e d 2 2 ϕ Eab ≈ Ea + Eb + 3 · |hi|ai| · |hi|bi| , (17) 4π0rr -π/ ˜ ˜ ˜ ˜ -π where a ∈ {⇑1, ⇓1} and b ∈ {⇑2, ⇓2}. When one inte- grates the energies given by Eq. 17 to find α, β, γ, δ and get an expression for φ, the single-qubit energies (Ea and ( ) Eb in Eq. 17) cancel, and the remaining terms can be simplified to FIG. 10. The phase of a cphase gate after subtracting local phases as a function of gate duration. The infidelity due to Z T 2 2 nonadiabaticity is always below 10−3 and could be decreased −e d 2 2 φ ≈ dt · (|hi| ⇑˜ i| − |hi| ⇓˜ i| ) further by attenuating the driving field and increasing gate 4π r3 1 1 (18) 0 0 r time. A cz gate, with φ = π, is implemented when T = ˜ 2 ˜ 2 × (|hi| ⇑2i| − |hi| ⇓2i| ). 494 ns. A key first question is how the dipole-dipole interaction affects the qubits while idling. For our choice of in ωE. Like the Rz gates, we parameterize cphase idling states, both qubit states have the electron in the gates in terms of their total duration T and calculate 2 2 |gi state, so |hi| ⇑i|˜ = |hi| ⇓i|˜ . This implies that φ ≈ 0. the resulting angle φ. We choose ∆E(t) = ∆Eidle + Furthermore, even if only one qubit is idling, the corre- l(t, τ1, −∆Eidle + D, τ1 + τac, −∆Eidle + D,T ), Ea(t) = sponding factor in Eq. 18 will be zero, so there will still Emax ·w(t−τ1, τ2, τac), and ωE = ε0 +A/4−hAi/2−2π · −1 −1 be no accumulated φ. The fact that there is no entangle- 10 MHz, where ∆Eidle = 10 000 V m , D = 2000 V m ment when at least one of the qubits is idling is a major is the value ∆E moves to during the gate, Emax = advantage of this choice of idling point and basis states. (40 V m−1) · min(1, (T/300 ns)2) is the max amplitude Eq. 18 implies that both qubits’ two computational of Eac, τ1 = 5 ns is the setup time, τac = T − 2τ1 is states must have different average dipoles in order for the time when ∆E is constant and Ea is nonzero, and a nonzero φ to accumulate. We can achieve this with τ2 = min(300 ns, τac/2) is the ramp-up time of Ea. When only oscillating electric fields at each qubit. The key T is small, Emax is smaller to maintain adiabaticity, and ingredient is the fact that the two computational states the ramp-up time τ2 must also be small to fit within T . have different ground-excited electron orbital splittings These pulses are very similar in shape to those used for due to the hyperfine interaction: E|e↓⇑i − E|g↓⇑i ≈ ε0 − the sweep Rx gate, except that here Bac = 0 and ∆E is A/4 + hAi/2, and E|e↓⇓i − E|g↓⇓i ≈ ε0 + A/4 − hAi/2. constant for most of the duration of the gate. ˜ Driving Eac for both qubits near, for example, the |⇓i Figure 10 shows the value of φ of our cphase gate as state’s electron orbital transition energy but far detuned the gate duration varies. Including the local Z-rotations from the |⇑i˜ state’s will give the |⇓i˜ state a significant |ei to correct the local phases, a cz gate takes ∼ 500 ns and component, creating a difference in dipole and leading to an arbitrary cphase gate takes less than 750 ns, fast a nonzero φ according to Eq. 18. enough that hundreds or thousands of two-qubit gates To give an example of a square-pulse implementation can be implemented within the decoherence time. of this idea, we assume a qubit spacing of r = 500 nm, To be truly practical, this cphase gate scheme must −1 and setting Ea = 30 V m , ωE = ε0 + A/4 − hAi/2 + have a degree of charge-noise-resistance. This is a signif- −1 2π · 5 MHz, Bac = 0, ∆E = 2000 V m , while keeping icant challenge because of the requirement that the two the other parameters the same as before, we find from H0 states of each qubit have different but precisely known (see Appendix A) that |⇑i˜ ≈ 0.923|g ↓⇑i − 0.368|e ↓⇑i dipole moments. Charge noise δE will lead to a pertur- id and |⇓i˜ ≈ 0.711|g ↓⇓i − 0.703|e ↓⇓i; the |⇓i˜ state is bation of the form −(deδE/2~)τz , which, taken to first driven closer to its electron orbital resonance and thus order, will then perturb the energies of the two qubit has a greater |ei component. We have changed ∆E from states by different amounts. This change in the qubit en- idling because in Eq. 5, the oscillating electric field is ergy splitting causes significant dephasing even for small δE. Finding a way to make this entangling gate scheme attenuated by a coefficient of Vt/ε0, so decreasing |∆E| will allow a smaller driving electric field to achieve the or an alternative scheme noise-resistant remains an open same effect. We can now use Eq. 18 and the definitions problem and will be the subject of future work. of the {|gi , |ei} states to find that φ/T ≈ 2π · 1.9 MHz. This yields a cz gate, with φ = π, in ∼ 500 ns. Performing a similar gate with realistic, smooth pulses VI. CONCLUSION is slightly more complicated due to the need to vary the AC electric field adiabatically and to the fact that To conclude, we have introduced quantum control the ground-excited splitting is changed slightly by the schemes to implement fast high-fidelity single- and two- presence of a second qubit, requiring an adjustment qubit gates for 31P nuclear spin qubits in silicon. We pre- 10

˜ ˜ † sented protocols for implementing arbitrary Rz(θ) and pressions, noting that Hj = H−j: Rx(θ) single-qubit gates, which can be combined to make an arbitrary, noise-resistant single-qubit gate in under A B (t)γ H˜ = (S − iS )(I + iI ) − a n (I + iI ) 500 ns. For typical charge noise levels of 100 V m−1, our ωE 4 x y x y 4 x y procedure achieves fidelities over 99.99% for arbitrary hAi A + − (S − iS )(I + iI ) Z-rotations and fidelities over 99.9% for arbitrary X- 2 4 x y x y rotations. This is well above the threshold error rate AVt VtB0γe∆γ of some quantum error correction codes, e.g. the sur- − (σx − iσy)SzIz + (σx − iσy)Sz 4ε0 4ε0 face code [30, 31]. We choose a computational basis such 2 2 that two qubits are only entangled when both are driven d e ∆EEa(t) − 2 σz. simultaneously, allowing for single-qubit gates to be per- 4~ ε0 formed without crosstalk from adjacent idling qubits. We ˜ AVt H2ωE = − (σx − iσy)(Sx − iSy)(Ix + iIy) also introduced a method for implementing two-qubit 8ε0 controlled-phase gates with arbitrary phases that take V deE (t) − t a (σ − iσ ), less than 750 ns for an inter-qubit distance of 0.5 µm, us- 8 ε x y ing only an oscillating electric field. These results are ~ 0 B (t)γ immediately relevant to ongoing experiments on donor- ˜ a e H2ωB = (Sx − iSy), based nuclear spin qubits. 4 B (t)γ H˜ = − a n (I − iI ). 2ωB −ωE 4 x y (A1) A system with one driving frequency can be analyzed in a time-independent way by considering a Floquet Hamiltonian [32], an infinite Hamiltonian whose basis is ACKNOWLEDGMENTS the tensor product of the original basis with the space of integers, with each integer representing a Fourier component of the solution. When the RWA fails because This work is supported by the Army Research Office the driving is strong, perturbation theory on the Floquet (W911NF-17-0287). Hamiltonian can derive corrections (like in the Bloch- Siegert shift) that preserve the time-independent approximation. We do a similar process here, using the multi- frequency Floquet formalism given in Ref. 33. First, we construct the multi-frequency Floquet Hamil- tonian HF . Because the base Hamiltonian is 8×8 and there are 9 distinct frequencies, the Floquet Hamilto- Appendix A: Multi-Frequency Floquet Theory nian will be 72×72. The diagonal will be populated with copies of H˜0 shifted by unique frequencies - just the frequencies present in H˜ in our truncated approximation - This appendix shows how to use multi-frequency Flo- and the off-diagonal will have the components of H˜ whose quet theory and second-order perturbation theory to frequencies are the differences of the shifts of the matri- find H0, an accurate time-independent approximation ces along the diagonal. H , truncated to one order in to the system Hamiltonian. The first step is to find F each frequency, is shown in Eq. A2, in which each matrix {H˜ } , the frequency components of H˜ such that ωj j element represents the corresponding 8 × 8 matrix. It ˜ P ˜ iωj t H = j Hωj e . The frequencies present are {ωi}i will turn out that only the matrices that are part of the = {0, ±ωE, ±2ωE, ±2ωB, ±(2ωB − ωE)}. H˜0 is given in diagonal or the central row or column will matter to our Section III, and the rest are given by the following ex- approximation, but we include them all for completeness. 11

 ˜ ˜  H0 − 2ωE 0 H−ωE 0 H−2ωE 0 0 0 0  0 H˜ − 2ω H˜ H˜ H˜ 0 0 0 0   0 B −2ωB +ωE −ωE −2ωE   H˜ H˜ H˜ − ω 0 H˜ H˜ H˜ 0 0   ωE 2ωB −ωE 0 E −ωE −2ωB −2ωB   0 H 0 H˜ − 2ω + ω H˜ 0 H˜ 0 0   ωE 0 B E −2ωE +ωE −2ωB  H =  H˜ H˜ H˜ H˜ H˜ H˜ H˜ H˜ H˜  F  2ωE 2ωB ωE 2ωB −ωE 0 −2ωB +ωE ωE 2ωB 2ωE   0 0 H˜ 0 H˜ H˜ + 2ω − ω 0 H˜ 0   2ωB 2ωB −ωE 0 B E ωE   ˜ ˜ ˜ ˜ ˜ ˜   0 0 H2ωE H2ωB HωE 0 H0 + ωE H−2ωB +ωE H−ωE   ˜ ˜ ˜ ˜   0 0 0 0 H2ωB HωE H2ωB −ωE H + 2ωB 0  ˜ ˜ ˜ 0 0 0 0 H2ωE 0 HωE 0 H0 + 2ωE, (A2)

If we note that the eigenvalues of H˜0 will be much derivative of the qubit energy difference with respect to smaller than either ωE or ωB (by a factor of over ∼ 10 the electric field, with our parameters), the dynamics can be easily ap- 2 proximated. The matrices along the diagonal are all very dδq AdeVt ˜ = − 3 . (B1) well-separated in energy from H0, so we can treat all the d∆E 4~ε0 off-diagonal elements in HF as a perturbation and derive an effective H0 using second-order quasi-degenerate When de∆E/~ Vt, ε0 ≈ de∆E/~, so the above equa- perturbation theory, also called a Schrieffer-Wolff trans- tion simplifies to formation [34]. 2 2 We choose the 8-dimensional subspace through the un- dδq A~ Vt ≈ − 2 2 3 . (B2) shifted H˜0 at the center of HF as the target subspace d∆E 4d e ∆E of the Schrieffer-Wolff transformation. We now define (0) With our idling parameters, this is 2π · 70 Hz/Vm−1, HF as the diagonal (the diagonal, not just a block 0 so assuming a typical noise in ∆E of 100 V m−1 gives a diagonal) of HF , and define the perturbation HF so (0) 0 dephasing time on the order of 0.1 ms. HF = HF + HF . Our goal is to apply a small transformation such that the target subspace becomes decoupled from the rest of the Floquet Hamiltonian, leaving an ef- Appendix C: Effect of a Time-Dependent {|gi , |ei} fective Hamiltonian H . As derived in Ref. 34, this is eff Basis given by

(0) (1) (2) When we transform the system Hamiltonian from the Heff = H + H + H + ..., (A3) (|ii, |di) basis to the (|gi, |ei) basis in Section II, we treat where ∆E as static and don’t include a −iΛΛ˙ † term for that change of basis, even though ∆E is not constant during (0) ˜0 Hmm0 = HF mm0 , gates. Here we give that correction and show that it is (1) 0 negligible, a conclusion which we checked by comparing H 0 = H , mm F mm0 our simulations to simulations in the original (|ii, |di) " # (2) 1 X 0 0 1 1 basis. Hmm0 = HF mlHF lm0 + , 2 Em − El Em0 − El The transformation from the (|ii, |di) basis to the (|gi, l |ei) is given by (A4) where the states m and m0 are states within the tar- 1 r de∆E i r de∆E get subspace and El is the energy of state l before the Λ = √ 1 + 1 − √ 1 − σy, (C1) 0 0 2 ~ε0 2 ~ε0 perturbation, which is simply HF ll because HF is di- (0) ˜ agonal. Note that H is just the diagonal of H0 and yielding (1) (0) (1) H is the off-diagonal, so H + H = H˜0. Summing the 0th-, 1st- and 2nd-order terms in H gives H0, the eff ˙ † deVt d∆E accurate, time-independent approximation mentioned in − iΛΛ = 2 σy · . (C2) 2~ε0 dt Section III, with typical fidelity > .9999 to the exact evolution for our gates. 2 The factor deVt/2~ε0 reaches a maximum of ∼ 2 × 10−3 m V−1 when ∆E = 0. There are two situations where this extra term could be Appendix B: Qubit Dephasing Rates problematic. First, during the very rapid shifts in electric field at the start and end of the Z- and X-rotations, the We here derive the dephasing rate of the qubit in our large term could cause unwanted |gi − |ei coupling. Sec- scheme due to charge noise. The critical value is the ond, in the middle of an X-rotation, the system dynamics 12 are fairly sensitive to the detunings between states, so a it does not significantly affect the gate. smaller added term from the more-slowly changing electric field could be problematic. In the second case, in the middle of an X-rotation, the For the first case, at maximum, |d∆E/dt| ∼ term is of order 2π · 5 MHz. This is two orders of magni- 1013 V m−1 s−1, so Eq. C2 is of order 2π · 1 GHz, while tude smaller than the detunings and much smaller than ε0 ∼ 2π · 5 GHz. This term is big enough to be problem- the energy splitting ε0 of the states it couples. If it were atic if it were turned on suddenly, but the fact that it included in the derivation of H0, it would only contribute waxes and wanes over the course of several ns preserves as a small correction. We have confirmed that it does not adiabaticity, and it is only large for a very short time, so significantly affect the results of our simulations.

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