Realistic system

In this section, we analyze the dominant factors degrading the gate delity in the realistic system of quantum-dot electron spin qubits in isotopically puried silicon, which include the electrical noise βU −(t), the uncertainty αt0 in tunnel coupling t0, and the ltering eects on the control pulses due to the nite bandwidth of waveform generator [83, 84].

Therefore, a more realistic Hamiltonian taking these factors into account becomes

H(t) = h

where Ω^filt_X (t) and Ω^filt_Y (t)are the actual output eld amplitudes with the ltering eects accounted for.

To understand the inuence of the electrical noise βU −(t)on the dynamics of the qubits, we simulate the experiment in Section 7 of the Supplementary Information of the paper by Veldhorst et al. [72]. In this experiment, the probability of the state |↑↓i, P (|↑↓i), is measured after the operations (π/2)X2 →(π/2)_Z2 →C-phase(τZ) → (π/2)Y2 with initial state |↓, ↓i for dierent τZ (gate operation time of C-phase gate). Gates (π/2)X2, (π/2)Y2, and (π/2)Z2 represent π

2 rotation in X-direction, Y-direction, and Z-direction, respectively, for dot2 qubit, and identity operation for dot1 qubit simultaneously. In experiment, C-phase is realized by tuning  up to a large constant value (small U − , U > ) for a period time τZ, and then tuning  down to a small constant value (large U − ) to turn o the two-qubit coupling. To see the probability oscillation from the two-qubit coupling, the probability in experiment is measured in the rotating frame by

U₁(t) =

to eliminate EZand δEZin the Hamiltonian. If there is no electrical noise, this probability

P (|↑, ↓i) ∼= 1 (Schrieer-Wol transformation with approximation in Sec. 3.2). For the realistic case (taking the electrical noise βU −(t) into consideration), we use the realistic Hamiltonian in Eq. (3.22) with E_X^filt(t) = 0, αt0 = 0, EZ = 39.16GHz, δEZ = −40MHz, t0 = 900MHz, and U −  = 300GHz, and the electrical noise βU −(t), chosen as a static and stochastic noise model (i.e., the noise strength is a time-independent constant value in each single run of experiment but this constant value can stochastically vary for dierent runs) with noise strengths obeying a normal distribution with standard deviation σU − and mean value µU − to simulate the C-phase gate suering the electrical noise eect. We assume (π/2)_X2, (π/2)Z2, and (π/2)Y2 are all ideal gates. The ensemble average probability (1000 dierent βU −(t) noise realizations), hP (|↑↓i)i, is calculated for σ = 0, 3, 10GHz (mean value µU −= 0) in Figure 3.6. One observes that hP (|↑↓i)i simulation with σU − = 3GHz is very close to the experiment in FIG. S6 in Section 7 of the Supplementary Information [72] as shown in Figure 3.7. Therefore, we assume the electrical noise βU −(t)is static and stochastic with standard deviation σU − = 3GHz for the rest of quantum gate operation simulations.

In experiment, the interdot tunnel coupling t0 is obtained by tting the experimental data with the simulation by the ideal Hamiltonian in Eq. (3.1). Therefore, there may exist some uncertainty value αt0 for t0 extraction. We regard αt0 as a systematic error, that is αt0 is a xed constant value for a specic two-qubit system, but the xed constant αt0

can vary for dierent two-qubit systems. For ideal system, U −  and t0 are converted

0 1 2 3 4 5 6 7 8 0

0.2 0.4 0.6 0.8 1

Figure 3.6.: Ensemble average probability hP (|↑↓i)i simulation for stochastic and static electrical noise βU − with mean value µU − = 0 and with standard deviation σ_{U −} = 0, 3, 10GHz.

Figure 3.7.: The experimentally measured hP (|↑↓i)i (courtesy of Veldhorst et al., 2015).

to Jp and Jm (Jp ≈ J_m ∼= t²₀

U − ) in ˜H^SWA_I,4×4(t) in Eq. (3.8) by SWA. For the realistic system, the uncertainty αt0 and the electrical noise βU −are accompanied by t0 and U − , respectively, into Jp and Jm in ˜H_I,4×4^SWA(t)as the form Jp ≈ J_m ∼= (t0+ αt0)²

U −  + β_{U −}. Assume αt0 and βU − are small uctuations compared with t0 and U − , respectively, and then Jp ≈ J_m ∼= t²₀

U −  + 2t0

U − αt0 − t²₀

(U − )²βU −. So 2t0

U − αt0 and − t²₀

(U − )²βU − are the corresponding uncertainty and noise contributions in the Hamiltonian, and the most important is that both uncertainty and noise contributions appear in the same locations of the Hamiltonian ˜H^SWA_I,4×4(t). Therefore, once the static electrical noise βU − is suppressed, and the uncertainty error αt0 is also minimized. Next, we simulate J2,t0 versus hJ2,U −i, the lowest-order contribution to the ensemble average indelity hIi for the systematic error αt0 and the stochastic and static noise βU −, respectively, from an ensemble of J1

optimized control parameter sets of the CNOT gate (EZ = 39.16GHz, δEZ = −40MHz, t0 = 900MHz, U −  = 300GHz, and tf = 500ns) as shown in Figure 3.8. One can see that as hJ2,U −i decreases J2,t0 also decreases with a constant ratio. By estimation, this ratio

10^-5 10^-4 10^-3 10^-2 10^-5

10^-4 10^-3 10^-2

Figure 3.8.: J2,t0 versus hJ2,U −i with σU −= 3GHz for αt0 = 9MHz and 4.5MHz.

J_2,t0 hJ_{2,U −}i

∼=

 2t0

U − αt0 · t_f

2

 t²₀

(U − )²σU −· t_f

2 = 4(U − )² t²₀

 αt0

σU −

2

(3.26)

with σU − = 3GHz is ∼ 4 and ∼ 1 for αt0 = 9MHz and 4.5MHz, respectively. These estimated ratios are comparable with our simulation in Figures 3.8. Therefore, J2,t0 can be simultaneously minimized when we suppress hJ2,U −i only under the conditions of the realistic values of system parameters and the electrical noise model. In other conditions, J_2,t0 may not linearly correlate to hJ2,U −i, and we should include both J2,t0 and hJ2,U −i in the cost function hJ2i for optimization.

We choose the form of the control pulses as

ΩX(t) =

kmax

X

k=1

aksin³(ωX,k· t) , (3.27)

ΩY(t) =

kmax

X

k=1

bksin³(ωY,k· t) , (3.28)

to construct the optimal CNOT gates and all optimal single-qubit gates of the system.

Here

ω_X,k= (2k − 1)π

t_f , (3.29)

ωY,k = (2k)π

t_f , (3.30)

and {a1, a₂, · · · , a_kmax} and {b1, b₂, · · · , b_kmax} are the control parameter sets for ΩX(t) and ΩY(t), respectively. By using the third power of sine function with the oscillation frequency ωX,k and ωY,k in Eq. (3.29) and Eq. (3.30) to compose the pulse, ΩX(t) and ΩY(t)have zero pulse strength and zero pulse slope at t = 0 and t = tf, which guarantees the smooth pulse-pulse connection of adjacent gates to reduce the extra error from the rise time issue. Besides, for the controllability of the quantum gates of this system, we should choose the symmetric form (symmetric to the middle gate operation time tf/2) for pulse ΩX(t)by ωX,k in Eq. (3.29), and antisymmetric form for pulse ΩY(t)by ωY,k in Eq.

(3.30). In experiment, there exist some realistic constraints on the control pulses such as the limitation of the maximum pulse strength and the ltering eects. For the power limitation of the on-chip ESR line, the maximum strength of both control pulses |ΩX(t)|and |ΩY(t)|

is limited by 1mT. This realistic constraint limits the region for searching the optimal control parameter set in the control parameter space {a1, a2, · · · , akmax, b1, b2, · · · , bkmax}, and thus the performance of the optimal gate we nd in the limited searching region could not be as good as that in the searching region without any constraints. Next, we discuss the ltering eects. When we input our optimal pulse Ω(t) (we use Ω(t) to represents Ω_X(t)and ΩY(t)pulses) to the instrument of the experiment, we expect the realistic pulse on the qubits is the same as our input. However, due to the nite bandwidth of waveform generators, our input optimal pulse Ω(t) is altered to the realistic ltered pulse Ω^{f ilt}(t)via the transfer function [83, 84]

Ω^filt(t) = 1 2π

ˆ _+∞

−∞

dt⁰ ˆ _+∞

−∞

dωF (ω)e^i(t−t0)ωΩ(t⁰), (3.31)

where

F (ω) = exp(−ω²/ω²₀) (3.32)

is the response function of the lter with ω0 being the cuto frequency. The transfer function can be rewritten as

Ω^filt(t) = 1 2π

ˆ _+∞

−∞

dωe^iωtF (ω)Ω(ω), (3.33)

where

Ω(ω) = ˆ _+∞

−∞

dt⁰e^−iωt0Ω(t⁰) (3.34) is the input optimal pulse in the frequency domain. Assume Ω(ω) distributes in the frequency region [−ω⁰, ω⁰]. If ω⁰is far below the cuto frequency ω0, the response function of the lter F (ω), dened in Eq. (3.32), in the preset frequency region [−ω⁰, ω⁰]approximates to 1, and thus the transfer function in Eq. (3.33) becomes Ω^filt(t) ∼= 1

2π

´_+ω⁰

−ω⁰ dωe^iωtΩ(ω), just the Fourier transformation of Ω(ω), and Ω^filt(t) ∼= Ω(t) . For this case, F (ω) in the transfer function doesn't work, and thus the ltering eects can be neglected. However, as ω⁰ approaches the cuto frequency ω0, F (ω) works by nullifying more and more high-frequency distribution of Ω(ω) in the transfer function, and thus the ltering eects become more and more apparent. Therefore, in the optimization process, we can not choose ωX,k

and ωY,k in the ΩX(t) and ΩY(t) as high as we need because F (ω) will nullify the high frequency component of the pulse, which is in the working region of F (ω). In Figure 3.9, we show the optimal pulses of CNOT gate with tf = 500ns in the frequency domain, ΩX(ω) and ΩY(ω). One can see that most frequency distribution of ΩX(ω) and ΩY(ω) is around and below ∼ 20MHz because we choose the number of control parameters kmax= 11 for both ΩX(t)and ΩY(t), and the maximum ωX,kmax/2π = 21MHzand ωY,kmax/2π = 22MHz by Eq. (3.29) and Eq. (3.30). But there still exist some distribution of ΩX(ω) and ΩY(ω) in the frequency higher than ∼ 20MHz. This is because we choose the third power of sine function to compose the pulse as shown in Eq. (3.27) and Eq. (3.28), and the third power of sine function can be expanded to the rst power of sine function, i.e., sin³(ω_X,k· t) =

3

4sin (ωX,k· t) − ¹₄sin (3ωX,k· t), and the contribution of the higher-frequency distribution of ΩX(ω) and ΩY(ω) just comes from sin (3ωX,k· t) and sin (3ωY,k· t), respectively. Next, we show the ltering eects on the optimal pulses of the CNOT gate in Figure 3.9 by the transfer function in Eq. (3.33) with the response function of the lter F (ω) in Eq.

(3.32). We vary the cuto frequency ω0/2π from 425.4MHz to 50MHz to see the pulse shift as shown in Figure. 3.10 and the corresponding indelity J1and hJ2,U −idegradation

0 20 40 60 80 100 -100

-50 0 50 100

0 20 40 60 80 100

-100 -50 0 50 100

Figure 3.9.: ΩX(ω) and ΩY(ω)of the optimal CNOT gate with tf = 500ns.

as shown in Figure 3.11. One can see that as ω0 decreases the pulse shift is more and more apparent and the corresponding indelity J1 is getting worse and worse, but the corresponding hJ2,U −i is not sensitive to ω0 until ω0/2π ≤ 100MHz, which implies that the hJ2,U −itopography in the control parameter space {a1, a₂, · · · , a_kmax, b₁, b₂, · · · , b_kmax} around our optimal control parameter set is very at. Therefore, we can add a ne-tuning optimization (substituting the ltered pulse Ω^filt(t)into the cost functions) after the two-step optimization introduced in Sec. 2.2 to recover the J1 degradation and keep hJ2,U −i unchanged. We use the assumption of ω0/2π = 425.4MHz (approximation for Tektronix AWG5014 [83]) for simulating the ltering eects on the quantum gates we demonstrate in the following section.