# Devitalizing noise-driven instability of entangling logic in silicon devices
with bias controls

Hoon Ryu  Ji-Hoon Kang  Division of National Supercomputing,
Korea Institute of Science and Technology Information, Daejeon 34141, Republic
of Korea

###### Abstract

The quality of quantum bits (qubits) in silicon is highly vulnerable to charge
noise that is omni-present in semiconductor devices and is in principle hard
to be suppressed. For a realistically sized quantum dot system based on a
silicon-germanium heterostructure whose confinement is manipulated with
electrical biases imposed on top electrodes, we computationally explore the
noise-robustness of 2-qubit entangling operations with a focus on the
controlled-X (CNOT) logic that is essential for designs of gate-based
universal quantum logic circuits. With device simulations based on the physics
of bulk semiconductors augmented with electronic structure calculations, we
not only quantify the degradation in fidelity of single-step CNOT operations
with respect to the strength of charge noise, but also discuss a strategy of
device engineering that can significantly enhance noise-robustness of CNOT
operations with almost no sacrifice of speed compared to the single-step case.
Details of device designs and controls that this work presents can establish a
rare but practical guideline for potential efforts to secure silicon-based
quantum processors using an electrode-driven quantum dot platform.

###### keywords:

Noise-robust entangling operations , Electrode-driven quantum dot structures ,
Silicon-based quantum computing , Computational nanoelectronics

##  1 Introduction

The spin of electrons in isotopically enriched silicon (Si) has been regarded
as a promising mechanism for encoding quantum information due to its extremely
long coherence time [1, 2, 3, 4] that is highly advantageous for stable
manipulations of quantum bits (qubits). In particular, a great amount of
effort has been put in by researchers to physically realize universal logic
gate devices with electron spins in Si-based quantum dot (QD) structures [3,
4, 5, 6, 7, 8, 9, 10, 11] whose confinement is controlled with external
electric and magnetic fields. The preciseness of corresponding logic
operations has been continuously improved so single qubit rotations can be now
conducted with a fidelity larger than 99% [3, 4, 5, 6, 7, 11], and recently a
98% fidelity is reported for 2-qubit SWAP operations [8]. Elaborated designs
of gate devices that generate quantum entanglement [12, 13], the most
celebrated quantum resource being widely used in various applications [14, 15,
16], have been also reported but their accuracy so far is generally not as
good as the non-entangling cases so the fidelity of 2-qubit Bell-states
generated from double quantum dot (DQD) platforms stays in 78%-89% [9, 10,
11]. With a rapid progress in pulsing technologies [6, 17], the speed of DQD-
based gating operations reached a sub-microsecond level, and it is shown that
a controlled-X (CNOT) operation, the most crucial entangling logic for
universal quantum computing, can be conducted in less than 200 nanoseconds
(nsec) with a single microwave pulse [10].

In general, the quality of spin qubits in solid-based platforms highly depends
on material-inherent noises [18, 19, 20, 21, 22] that are mainly due to the
fluctuation in local electric and magnetic fields around qubits. In the Si-
based case, noises of magnetic fields (spin noises) can be suppressed with
purification of 28Si crystals, and latest works have shown 12-inch 28Si wafers
that contain 100ppm or less of spin-carrying 29Si atoms [23, 24]. Suppressing
noises of electric fields (charge noises), however, is more difficult than the
case of spin noises since its origins have not been fully understood yet.
Accordingly, state-of-the-art ideas have been proposed to increase the
robustness of spin qubits to existing charge noises in Si devices such as, for
example, placing qubits far away from surface oxides [22] that can serve as a
source of low-frequency charge noises [19], increasing spin resonance
frequencies [7], and biasing DQDs symmetrically to reduce the sensitivity of
qubit interactions to charge noises [10, 25]. In spite of the non-trivial
contribution driven with these ideas, the up-to-date fidelity of entangling
operations in Si devices is not yet in a level where the accuracy in
computations can be generally guaranteed, and the motivation for sound studies
on technical strategies that can enhance the fidelity of entangling operations
under charge noises, therefore, should be huge.

In this work, we elaborately examine the engineering-driven possibilities for
devitalizing negative effects that charge noises have against entangling
operations implemented with Si QD devices, where the focal point of
engineering is the real-time pattern of control signals that has been rarely
discussed in detail by the strategies proposed in previous studies [7, 22,
25]. For this purpose, we computationally explore Si DQD structures with our
in-house simulation code package that can describe device operations in a
full-scale from initializations and time-dependent behaviors of electron spin
qubits. As a baseline for discussion, we first model the fast CNOT operation
that is driven with a single-step pulse in the recently reported DQD platform
[10], and quantify fluctuations in fidelity under charge noises by
incorporating random noisy potential profiles to device simulations. Then, as
an alternative way, we implement a CNOT operation with a multi-step control
that does not employ AC microwave pulses for generation of entanglement. In
spite of the loss in fidelity that happens during the real-time transition of
control signals, we find a general pattern that the resulting CNOT operation
has remarkably increased robustness to charge noises whilst its operating
speed can be maintained in a same order compared to the case of a single-step
control. Additional in-depth discussion is presented via rigorous modeling to
study the optimal control of multi-step CNOT operations in realistic
conditions with a trade-off between the speed and the noise-robustness of
operations. Being carried as an extension of our preliminary study that
focused on the noise-free addressing of individual qubits [26], this work can
make a meaningful contribution for Si-based designs of entangling logic blocks
that are essential for development of programmable quantum processors.

##  2 Methods

Figure 1(a) shows the DQD structure that is adopted as a target of modeling in
this work. Mimicking the reported physical system [10], the target platform is
based on a heterostructure that consists of 2 Si and 2 silicon-germanium
(SiGe) layers where the fraction of Ge in SiGe layers is 30%. Due to the
Si$/$SiGe band offset, the structure has a natural quantum well along the
vertical ([010]) direction and electrons can be confined in the 8
nanometer(nm)-thick Si layer. The lateral ([100]) confinement in the 8nm-thick
Si layer is controlled with DC biases imposed on top Ti$/$Au electrodes (2
barrier gate biases ($V_{B}$), 1 left$/$middle$/$right gate bias
($V_{L}$$/$$V_{M}$$/$$V_{R}$)), so the system can have up to 2 potential
valleys. As the DQD system is quite long ($>$100nm) along the [001] direction,
we use its 2D-slice as a simulation domain assuming the structure is
infinitely long along that direction. The top electrodes are considered in
device simulations by imposing a Dirichlet boundary condition on a 2D Poisson
equation with applied biases and Schottky barrier heights ($\Phi_{B}$) that
are calculated using the work-function reported for Ti/Au metal layers. [27]
The source and drain electron reservoirs, which are secured with 2D electron
gas (2DEGs) in reality, are also described with Dirichlet boundaries (the two
red boundaries in Figure 1(a)) where we set $\Phi_{B}$ to zero assuming that
2DEGs are formed well and are therefore perfectly ohmic. For simulations, we
grounded the source and imposed an extremely small bias ($\varepsilon$
$\simeq$ 0.1mV) on the drain, and a low temperature of 1.5K is assumed.

The spatial distribution of potential energy and electron density in the DQD
system, which is the outcome of device simulations, is determined with a self-
consistent process described in Figure 1(b). While the potential profile is
calculated with a normal Poisson solver, the charge profile is evaluated in
two ways with regional dependence so the region of thin Si layers (labeled as
Quantum Region), which has most of electrons and must be solved quantum
mechanically, is treated with electronic structure simulations coupled to a
parabolic effective mass model [28], and the region of SiGe layers (labeled as
Bulk Region) is solved with the physics of bulk semi-conductors. For precise
modeling of spin states, the electronic structure is calculated with a lateral
distribution of the static magnetic field along the [001] direction ($B_{Z}$)
that is reported by Neumann $et$ $al.$ [29] with simulations of the horseshoe-
shaped micromagnet employed in the real experiments [10, 30, 31] (see the
inset of Figure 1(a)). Once the potential distribution at a certain set of
biases is determined, we disturb this “clean” solution with a noisy potential
profile, which is obtained with values that are randomly generated per each
real-space grid of the simulation domain as described in Figure 2(a). All the
random values here are generated under a zero-mean gaussian distribution, and
its standard deviation $\sigma$, which represents the strength of charge
noises, is considered up to 5$\mu$eV that is normal in Si-based devices these
days [32, 33, 34, 35]. Once the ground states of two QDs are known from device
simulations, we can construct the Heisenberg 2-spin Hamiltonian with their
Zeeman-spitting energies and exchange interaction [36], and 2-qubit time
responses of the DQD system can be then calculated as described in Figure 2(b)
that shows the scheme of our full-stack modeling.

##  3 Results and Discussion

In any physical platforms, the first step for gating operations is to
initialize qubits so the system can be prepared for upcoming control pulses.
In the target DQD platform where a qubit state 0 and 1 are encoded to the
down-spin ($|$$\downarrow$$\rangle$) and the up-spin ($|$$\uparrow$$\rangle$)
ground state of a QD, respectively, initialization is done by manipulating
biases imposed on top electrodes such that the $|$$\downarrow$$\rangle$ state
in each QD is occupied with an electron. To quantify the range of biases that
can place the target system in the ($|$$\downarrow$$\rangle_{L}$,
$|$$\downarrow$$\rangle_{R}$) state (= $|$$\downarrow$$\downarrow$$\rangle$)
where the subscription $L$ and $R$ represent for the left and right QD,
respectively, we model the charge stability with device simulations, and
present the result in Figure 3(a) as a function of $V_{L}$ and $V_{R}$ at
$V_{M}$ = 400mV, where $V_{B}$ is fixed to 200mV. The stability diagram is
split to 4 regimes, and each one is identified with two numbers that represent
the electron population of each QD. With increasing $V_{L(R)}$, the ground
state of the left(right) QD shifts down in energy and is occupied with an
electron when the state touches the Fermi-level of the source electron
reservoir. Establishing a strong connection to data measured for the physical
DQD system [10], our result reveals that ($V_{L}$, $V_{R}$) = (540mV, 570mV)
(the yellow point labeled as Pinit) can be an initialization point that is
beneficial for noise-robust qubit interactions since two QDs can be
symmetrically biased [25].

Representing the strength of inter-QD qubit interaction, the exchange energy
($J$) between two ground states serves as a source of 2-qubit entanglement in
the DQD platform and can be controlled with the middle gate bias that affects
the potential barrier between two QDs. In our simulations, $J$ becomes 75.6KHz
at $V_{M}$ = 400mV (at Pinit) and, as shown in Figure 3(b), sharply reaches
19.3MHz when $V_{M}$ is increased by 8mV. Changes in $V_{M}$ also affect
Zeeman-splitting energies of the left ($E_{ZL}$) and the right ground state
($E_{ZR}$) that determine the resonance frequency of each spin qubit, but
their dependence on $V_{M}$ is not quite noticeable such that ($E_{ZL}$,
$E_{ZR}$) is (18.309GHz, 18.453GHz) at $V_{M}$ = 400mV and changes to
(18.312GHz, 18.448GHz) when $V_{M}$ is 408mV. Due to the laterally
inhomogeneous $B_{Z}$ (the inset of Figure 1(a)), $E_{ZL}$ and $E_{ZR}$ are
distinguishable and qubits can be addressed independently if their interaction
is weak, and one of such cases is shown in Figure 3(c), where we simulated
2-qubit responses at $V_{M}$ = 400mV with a [010]-oriented time-varying
magnetic field $B_{Y}$${(t)}$ = $B_{o}$${cos(\omega_{D}t+\theta)}$ that is
generated from a microwave pulse and is incorporated in modeling as elements
of the Heisenberg Hamiltonian. In particular, the two subfigures here show
that a $R_{Y}(\pi)$ operation (1-qubit rotation by $\pi$ radian around the
Y-axis) can be selectively implemented with each qubit by setting $\omega_{D}$
to $E_{ZL}$ or $E_{ZR}$, and gating is completed in 99.34 & 99.47nsec (left &
right) when $B_{o}$ = 5.0MHz and $\theta$ = 0. If the interaction is not weak
enough to ignore, the resonance frequency of one qubit starts to depend on the
spin state ($|$$\downarrow$$\rangle$ or $|$$\uparrow$$\rangle$) of the other
qubit, and a CNOT operation can be then realized with a single control pulse
[10, 36]. To mimic this 1-step implementation with modeling, we simulate the
DQD structure at $V_{M}$ = 408mV with $B_{Y}$${(t)}$ of $\omega_{D}$ (=
1.832GHz), $B_{o}$ (= 4.977MHz) and $\theta$ (= 1.5$\pi$ radian) that are
determined with the analytical solution driven by Russ $et$ $al.$ [36].
Simulated 2-qubit responses in Figure 3(d) clearly show the CNOT gating is
completed in 100.4nsec, being fairly connected to the experiment [10].

In a noise-free condition, modeling results show that the $R_{Y}(\pi)$
operation is conducted for the left and the right spin with a fidelity of
99.93% and 99.98%, respectively, and the 1-step CNOT operation has a fidelity
of 98.34%. To investigate how they are affected by charge noises, we simulate
the system with the conditions described in the previous paragraph but disturb
the clean potential profiles with random noisy values that are generated under
a zero-mean gaussian distribution of a standard deviation $\sigma$. Figure
4(a) and 4(b) show the fidelity of the $R_{Y}(\pi)$ and the 1-step CNOT
operation as a function of $\sigma$, respectively, where each case is modeled
by conducting 1,000 simulations per a single value of $\sigma$ that is varied
from 10-3 to 5$\mu$eV. Results clearly indicate that both operations continue
to lose accuracy as the DQD system experiences more severe noises, but their
patterns of the noise-driven degradation in fidelity are different. In the
case of $R_{Y}(\pi)$ gating, the fidelity turns out to be 99.93$\pm$10-6%
(left) and 99.98$\pm$10-6% (right) at $\sigma$ = $10^{-3}$$\mu$eV, and starts
to decrease noticeably when $\sigma$ reaches 1$\mu$eV or larger such that it
drops to 96.95$\pm$0.5664% (left) and 96.97$\pm$0.5687% (right) at $\sigma$ =
5$\mu$eV, Similarly to the $R_{Y}(\pi)$ case, the 1-step CNOT operation has a
nice fidelity (98.34$\pm$0.003%) when $\sigma$ is $10^{-3}$$\mu$eV. Its
robustness to noises however is much worse than what $R_{Y}(\pi)$ shows, and
the average fidelity plummets more than 60% (32.84$\pm$0.5361%) when $\sigma$
is 5$\mu$eV.

In the extreme case where the left and right qubit never interact ($i.e.$, $J$
= 0), the Heisenberg 2-spin Hamiltonian can be completely described with
Zeeman-splitting energies of spin states and external magnetic fields, and so
are 2-qubit responses of the DQD system. Accordingly, in the regime of a weak
interactions that can be represented with the case of $V_{M}$ = 400mV, the
quality of single qubit addressing under charge noises should be determined by
how $E_{ZL}$ and $E_{ZR}$ behave. The origin of noise-robust $R_{Y}(\pi)$
rotations (Figure 4(a)) can be therefore clarified with simulation results
presented in the upper subfigure of Figure 4(c), which indicate that the
noise-driven fluctuation in two Zeeman-splitting energies at $V_{M}$ = 400mV
becomes smaller than 100Hz (10-5% of their clean values) regardless of
$\sigma$. If the 2-qubit interaction is not ignorable as it is when $V_{M}$ =
408mV, $J$ also starts to affect the noise-robustness of gating. As Figure
4(d) shows, the noise-driven fluctuation in $J$ is generally much stronger
than the $E_{ZL}$ & $E_{ZR}$ case, and, particularly at $V_{M}$ = 408mV, it
acts as the major factor that determines the noise-robustness of 2-qubit
states because our results reveal that the fluctuation in $E_{ZL}$ and
$E_{ZR}$ is still negligible as shown in the lower subfigure of Figure 4(c).
In consequence, we can say that the huge reduction in fidelity observed in
noisy 1-step CNOT operations (Figure 4(b)) is mainly due to the noise-driven
instability of $J$.

Given that the material-inherent charge noise itself would not be easy to be
eliminated or hugely suppressed, the next action for implementation of
reliable CNOT operations may be to seek for engineering approaches that can
make the gate more robust to “existing” noises. For this purpose, here we
computationally explore one idea whose main focus is to control qubit
interactions such that the “noise-sensitive” interval in time responses can be
reduced as much as possible. In Figure 5(a)-(i), we show a simple 2-qubit
circuit which also conducts a CNOT operation and will be used as a testbed of
the noise-robustness. Here, the desired gating can be implemented with a time-
sequential conduction of a $R_{Y}(-\pi/2)$, a controlled-Z (CZ), and a
$R_{Y}(\pi/2)$ operation where $R_{Y}$ rotations are applied to the upper
(target) qubit. Taking the right QD spin as a control qubit, we can implement
the two $R_{Y}$ gates in the DQD platform at $V_{M}$ = 400mV by setting
$B_{Y}$${(t)}$ similarly to the $R_{Y}(\pi)$ case except that $\theta$ is
$\pi$ (instead of 0) when the rotation angle is negative. The CZ gate in the
second step serves as an entangling block and can be obtained without time-
varying microwave pulses. Technically, a CZ gate can be further decomposed
into 2 steps as illustrated in the bottom subfigure of Figure 5(a)-(i). Here,
the 2-qubit controlled-phase gate $U$ is $device$-$native$ [36], which means
the unitary can be completely described with only DQD-native spin parameters
($i.e.$, Zeeman-splitting energies and exchange interaction). The Z-rotation
($R_{Z}$) is also device-native but must be carried in the regime of a weak
interaction ($e.g.$, $V_{M}$ = 400mV in our case). In real experiments, the
$R_{Z}$ is conducted by changing the reference phase for individual spins
instead of directly rotating them, which can be done conveniently with
software at negligible cost in speed and accuracy [7, 9, 10]. In Figure
5(a)-(ii), we show the real-time pattern of $V_{M}$ that drives this multi-
step CNOT gate, where $\tau_{Y}$’s and $\tau_{U}$ on the X-axis are the gating
time of $R_{Y}(\pm\pi/2)$ and $U$, respectively. We assume that the $R_{Z}$
gating is performed instantaneously (at the time point labeled as
$\bf{T_{Z}}$), adopting a bias-transition time ($\tau_{TR}$) of 5nsec for
simulations similarly to the experiment [10]. The resulting responses in
Figure 5(a)-(iii) reveal that the entire process takes 132.1nsec, where
$\tau_{Y}(\pm\pi/2)$ and $\tau_{U}$ become 48.1nsec and 25.9nsec,
respectively.

The focal characteristic of the above-mentioned multi-step CNOT operation is
that the 2-qubit entanglement is solely generated by the CZ block, and
eventually by the controlled-phase unitary $U$, as all the remaining logics
($R_{Y}$’s and $R_{Z}$’s) handle 1-qubit addressing in the regime of a weak
interaction. As the sensitivity of $E_{ZL}$ and $E_{ZR}$ to charge noises is
not quite noticeable (Figure 4(c)), the fairly nice noise-robustness of
$R_{Y}$ shown in Figure 4(a) also becomes valid for 1-qubit rotations about
arbitrary axes. We can thus expect that the noise-driven fidelity of the CZ
operation may strongly depend on that of $U$, and this can be confirmed with
Figure 5(b) that shows the simulated pattern in fidelity of CZ and $U$ gating.
Due to the negligible role of $R_{Y}$ blocks, the overall fidelity of the
multi-step CNOT logic, shown with a red dotted line of square marks in Figure
5(c), also closely follows the fidelity of $U$. When $V_{M}$ is 408mV, the
multi-step CNOT logic generates 2-qubit entanglement in $\sim$4x less time
(25.9nsec) than the 1-step gating (100.4nsec). This “reduced time-period of a
strong interaction” can contribute to making the operation more robust to
charge noises, so the simulated fidelity of the multi-step operation at
$\sigma$ = 5$\mu$eV becomes 69.81$\pm$0.8208% while the 1-step CNOT gate shows
32.84$\pm$0.5361% in the same conduction. Our result in Figure 5(c) also
confirms the core message remains effective in the entire range of $\sigma$
that is considered for simulations.

In Figure 3(b), we showed that the interaction energy between QDs has little
effects on the resonance frequency of each spin qubit, so the gating time of
$U$ can be safely controlled with $J$ (and thus $V_{M}$) with no worries for
unintentional variations in any $E_{ZL}$\- and $E_{ZR}$-related elements of
the 4$\times$4 Heisenberg Hamiltonian [36]. With this background, we
investigate what happens on the noise-robustness of the multi-step CNOT
operation if the gating time of $U$ is further reduced. For this purpose, the
multi-step CNOT gate is simulated at $V_{M}$ = 410mV and 412mV, where other
control parameters are kept the same as the previously used ones. The time
responses in Figure 6(a) clearly show that the entanglement is generated in
7.2nsec and 1.9nsec when $V_{M}$ is 410mV and 412mV, respectively, and thus
the CNOT gating time is reduced to 113.4nsec and 108.1nsec. Figure 6(b), which
shows the fidelity of each noisy CNOT operation, indicates that the noise-
robustness at $V_{M}$ = 410mV does not quite change compared to the case of
$V_{M}$ = 408mV though entanglement is generated must faster (25.9nsec
$\rightarrow$ 7.2nsec). This result, being different from the one obtained
through a comparison between the single-step and the multi-step CNOT gate at
$V_{M}$ = 408mV, can be explained with the fact that the time-integration of
$J$ ($i.e.$, $\int_{0}^{\tau_{U}}J(t)\,dt$) remains the same in the two cases
(19.3MHz$\times$25.9nsec and 69.5MHz$\times$7.2nsec when $V_{M}$ is 408mV and
410mV, respectively), while, in the previous two cases where $V_{M}$ is kept
the same, the time-integration becomes smaller in the multi-step operation
(19.3MHz$\times$25.9nsec) than in the single-step one
(19.3MHz$\times$100.4nsec). If $V_{M}$ is increased to 412mV, the time-
integration still remains similar (266.1MHz$\times$1.9ns), showing $<$1%
deviation from the values at $V_{M}$ = 408mV and 410mV. In this case, however,
the average fidelity gets worse even under weak noises (75.1% at $\sigma$ =
10-3$\mu$eV), and this is due to the transition of $V_{M}$ that is essential
to switch the interaction strength of QDs. Figure 6(c), which shows the loss
in fidelity of the multi-step CNOT operation at $V_{M}$ = 412mV as a function
of $\tau_{TR}$, indicates that the loss can be reduced with a faster bias-
transition, and we observe that the fidelity is recovered back to 98.52% if
the transition can be conducted in 1nsec. Overall, it is fair to say that
increasing the speed of U gating has little effects on the fidelity under
charge noises, but still contributes to saving the gating time, so, at $V_{M}$
= 410mV where the fidelity is not yet quite affected by a 5nsec-transition of
$V_{M}$, the multi-step CNOT gate can be completed with just 10% larger time-
cost (113.4nsec) than the single-step gate (100.4nsec).

##  4 Conclusion

Entangling logic operations under charge noises are computationally
investigated in a silicon double quantum dot (DQD) system where quantum bits
(qubits) are encoded to the confined electron spins. Using a realistic DQD
platform based on a silicon$/$silicon-germanium (Si$/$SiGe) heterostructure,
we make a solid connection to the recent experimental work [10] where a fast
controlled-X (CNOT) gate has been implemented with a single-step control, but
also extend the modeling scope into noise-driven behaviors of the single-step
CNOT operation and 1-qubit rotations by incorporating random noisy potential
energies into device simulations. Though the 1-step implementation of a CNOT
gate in the Si DQD platform has opened the fundamental pathway for securing a
fast CNOT gate with simple controls, it severely suffers from charge noises
due to unintended fluctuations in the interaction energy between QDs, so its
fidelity reaches lower than 35% when the standard deviation of noisy potential
energies ($\sigma$) is 5$\mu$eV. In contrast, 1-qubit rotations are generally
quite robust to charge noises since the noisy fluctuation in potential
distributions hardly affects the resonance frequency of individual spins.
Employing a DQD-native controlled-phase operation can be remarkably helpful
for noise-robust implementation of a CNOT gate, because, at the same strength
of 2-qubit interaction, it generates quantum entanglement much faster than the
single-step CNOT operation. Although additional 1-qubit rotations need to be
conducted sequentially in time to complete the CNOT operation, they have
little effects on the noise-robustness, so the overall fidelity reaches
$\sim$70% at $\sigma$= 5$\mu$eV in spite of the increased complexity in device
controls associated with additional 1-qubit rotations. Another benefit of the
controlled-phase operation implemented in the DQD platform is that its speed
can be enhanced by increasing the strength of 2-qubit interaction with almost
no degradation in noise-robustness. In consequence, the associated CNOT gating
can be conducted as fast as the single-step operation. Being supported with
rigorous simulations, the engineering details discussed in this work can
contribute to elevating the current status of a Si QD platform for robust
designs of scalable quantum processors.

## Acknowledgements

This work has been supported by the Korea Institute of Science and Technology
Information (KISTI) institutional R&D program (K-22-L02-C09) and by the grant
from the Institute for Information & Communications Technology Promotion
(2019-0-00003) funded by the Korea government (MSIP). The NURION high
performance computing resource has been extensively utilized for simulations.

## References

  * [1] T. Kobayashi, J. Salfi, C. Chua, J. van der Heijden, M. House, D. Culcer, W. Hutchison, B. Johnson, J. McCallum, H. Riemann, N. Abrosimov, P. Becker, H. Pohl, M. Y. Simmons, S. Rogge, Engineering long spin coherence times of spin-orbit qubits in silicon, Nature Materials 20 (2021) 38–42.
  * [2] J. T. Muhonen, J. P. Dehollain, A. Laucht, F. E. Hudson, R. Kalra, T. Sekiguchi, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak, A. Morello, Storing quantum information for 30 seconds in a nanoelectronic device, Nature Nanotechnology 9 (2014) 986–991.
  * [3] M. Veldhorst, J. C. C. Hwang, C. H. Yang, A. W. Leenstra, B. de Ronde, J. P. Dehollain, J. T. Muhonen, F. E. Hudson, K. M. Itoh, A. Morello, A. S. Dzurak, An addressable quantum dot qubit with fault-tolerant control-fidelity, Nature Nanotechnology 9 (2014) 981–985.
  * [4] E. Kawakami, T. Jullien, P. Scarlino, D. R. Ward, D. E. Savage, M. G. Lagally, V. V. Dobrovitski, M. Friesen, S. N. Coppersmith, M. A. Eriksson, L. M. K. Vandersypen, Gate fidelity and coherence of an electron spin in an Si$/$SiGe quantum dot with micromagnet, Proceedings of the National Academy of Sciences of the United States of America 113 (2016) 11738–11743.
  * [5] E. Kawakami, P. Scarlino, D. R. Ward, F. R. Braakman, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, L. M. K. Vandersypen, Electrical control of a long-lived spin qubit in a Si$/$SiGe quantum dot, Nature Nanotechnology 9 (2014) 666–670.
  * [6] K. Takeda, J. Kamioka, T. Otsuka, J. Yoneda, T. Nakajima, M. R. Delbecq, S. Amaha, G. Allison, T. Kodera, S. Oda, S. Tarucha, A fault-tolerant addressable spin qubit in a natural silicon quantum dot, Science Advances 2 (2016) e1600694.
  * [7] J. Yoneda, K. Takeda, T. Otsuka, T. Nakajima, M. R. Delbecq, G. Allison, T. Honda, T. Kodera, S. Oda, Y. Hoshi, N. Usami, K. M. Itoh, S. Tarucha, A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9%, Nature Nanotechnology 13 (2018) 102–106.
  * [8] A. J. Sigillito, M. J. Gullans, L. F. Edge, M. Borselli, J. R. Petta, Coherent transfer of quantum information in a silicon double quantum dot using resonant SWAP gates, npj Quantum Information 5 (2019) 110.
  * [9] T. F. Watson, S. G. J. Philips, E. Kawakami, D. R. Ward, P. Scarlino, M. Veldhorst, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, L. M. K. Vandersypen, A programmable two-qubit quantum processor in silicon, Nature 555 (2018) 633–637.
  * [10] D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans, J. M. Taylor, G. Burkard, J. R. Petta, Resonantly driven CNOT gate for electron spins, Science 359 (2018) 439–442.
  * [11] W. Huang, C. H. Yang, K. W. Chan, T. Tanttu, B. Hensen, R. C. C. Leon, M. A. Fogarty, J. C. C. Hwang, F. E. Hudson, K. M. Itoh, A. Morello, A. Laucht, A. S. Dzurak, Fidelity benchmarks for two-qubit gates in silicon, Nature 569 (2019) 532–536.
  * [12] W. K. Wootters, W. S. Leng, Quantum Entanglement as a Quantifiable Resource, Philosophical Transactions of the Royal Society A 356 (1998) 1717–1731.
  * [13] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement, Reviews of Modern Physics 81 (2009) 865–942.
  * [14] C. H. Bennett, G. Brassard, C. Cr$\acute{e}$peau, R. Jozsa, A. Peres, W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolskyrosen channels, Physical Review Letters 70 (1993) 1895.
  * [15] P. W. Shor, Algorithms for quantum computation: discrete logarithms and factoring, in: Proceedings of the Annual Symposium on Foundations of Computer Science, 1994, pp. 124–134.  doi:10.1109/SFCS.1994.365700.
  * [16] D. Lachance-Quirion, S. Wolski, Y. Tabuchi, S. Kono, K. Usami, Y. Nakamura, Entanglement-based single-shot detection of a single magnon with a superconducting qubit, Science 367 (2020) 425–428.
  * [17] L. M. K. Vandersypen, I. L. Chuang, NMR techniques for quantum control and computation, Reviews of Modern Physics 76 (2005) 1037–1069.
  * [18] A. V. Kuhlmann, J. Houel, A. Ludwig, L. Greuter, D. Reuter, A. D. Wieck, M. Poggio, R. J. Warburton, Charge noise and spin noise in a semiconductor quantum device, Nature Physics 9 (2013) 570–575.
  * [19] E. J. Connors, J. Nelson, H. Qiao, L. F. Edge, J. M. Nichol, Low-frequency charge noise in Si$/$SiGe quantum dots, Physical Review B 100 (2019) 165305.
  * [20] C. D. Wilen, S. Abdullah, N. Kurinsky, C. Stanford, L. Cardani, G. $\acute{D}$Imperio, C. Tomei, L. Faoro, L. Loffe, C. Liu, A. Opremcak, B. Christensen, J. DuBois, R. McDermott, Correlated charge noise and relaxation errors in superconducting qubits, Nature 594 (2021) 369–373.
  * [21] S. Pezzagna, J. Meijera, Quantum computer based on color centers in diamond, Applied Physics Reviews 8 (2021) 011308.
  * [22] L. Kranz, S. K. Gorman, B. Thorgrimsson, Y. He, D. Keith, J. G. Keizer, M. Y. Simmons, Exploiting a Single-Crystal Environment to Minimize the Charge Noise on Qubits in Silicon, Advanced Materials 32 (2020) 2070298.
  * [23] R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohuslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, S. Franceschi, A CMOS silicon spin qubit, Nature Communications 7 (2016) 13575.
  * [24] V. Mazzocchi, P. Sennikov, A. Bulanov, M. Churbanov, B. Bertrand, L. Hutin, J. Barnes, M. Drozdov, J. Hartmann, M. Sanquer, 99.992% 28Si CVD-grown epilayer on 300mm substrates for large scale integration of silicon spin qubits, Journal of Crystal Growth 509 (2019) 1–7.
  * [25] M. Reed, B. Maune, R. Andrews, M. Borselli, K. Eng, M. Jura, A. Kiselev, T. Ladd, S. Merkel, I. Milosavljevic, E. Pritchett, M. Rakher, R. Ross, A. Schmitz, A. Smith, J. Wright, M. Gyure, A. Hunter, Reduced Sensitivity to Charge Noise in Semiconductor Spin Qubits via Symmetric Operation, Physical Review Letters 116 (2016) 110402.
  * [26] J. Kang, J. Ryu, H. Ryu, Exploring the behaviors of electrode-driven si quantum dot systems: from charge control to qubit operations, Nanoscale 13 (2021) 332–339.
  * [27] Y. Nosho, Y. Ohno, S. Kishimoto, T. Mizutani, Relation between conduction property and work function of contact metal in carbon nanotube field-effect transistors, Nanotechnology 17 (2006) 3412–3415.
  * [28] J. Wang, A. Rahman, A. Ghosh, G. Klimeck, M. Lundstrom, On the validity of the parabolic effective-mass approximation for the I-V calculation of silicon nanowire transistors, IEEE Transactions on Electron Devices 52 (2005) 1589–1595.
  * [29] R. Neumann, L. R. Schreiber, Simulation of micro-magnet stray-field dynamics for spin qubit manipulation, Journal of Applied Physics 117 (2015) 193903.
  * [30] J. Yoneda, T. Otsuka, T. Nakajima, T. Takakura, T. Obata, M. Pioro-Ladri$\grave{e}$re, H. Lu, C. J. Palmstr$\o$m, A. C. Gossard, S. Tarucha, Fast Electrical Control of Single Electron Spins in Quantum Dots with Vanishing Influence from Nuclear Spins, Physical Review Letters 113 (2014) 267601.
  * [31] J. Yoneda, T. Otsuka, T. Takakura, M. Pioro-Ladri$\grave{e}$re, R. Brunner, H. Lu, T. Nakajima, T. Obata, A. Noiri, C. J. Palmstr$\o$m, A. C. Gossard, S. Tarucha, Robust micromagnet design for fast electrical manipulations of single spins in quantum dots, Applied Physics Express 8 (2015) 084401.
  * [32] B. Thorgrimsson, D. Kim, Y.-C. Yang, L. W. Smith, C. B. Simmons, D. R. Ward, R. H. Foote, J. Corrigan, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, Extending the coherence of a quantum dot hybrid qubit, npj Quantum Information 3 (2017) 32.
  * [33] M. A. Fogarty, K. W. Chan, B. Hensen, W. Huang, T. Tanttu, C. H. Yang, A. Laucht, M. Veldhorst, F. E. Hudson, K. M. Itoh, D. Culcer, T. D. Ladd, A. Morello, A. S. Dzurak, Integrated silicon qubit platform with single-spin addressability, exchange control and single-shot singlet-triplet readout, Nature Communications 9 (2018) 4370.
  * [34] Z. Shi, C. B. Simmons, D. R. Ward, J. R. Prance, R. T. Mohr, T. S. Koh, J. K. Gamble, X. Wu, D. E. Savage, M. G. Lagally, S. N. C. M. Friesen, M. A. Eriksson, Coherent quantum oscillations and echo measurements of a Si charge qubit, Physical Review B 88 (2013) 075416.
  * [35] X. Wu, D. R. Ward, J. R. Prance, D. Kim, J. K. Gamble, R. T. Mohr, Z. Shi, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson, Two-axis control of a singlet-triplet qubit with an integrated micromagnet, Proceedings of the National Academy of Sciences of the United States of America 111 (2014) 11938–11942.
  * [36] M. Russ, D. M. Zajac, A. J. Sigillito, F. Borjans, J. M. Taylor, J. R. Petta, G. Burkard, High-fidelity quantum gates in Si/SiGe double quantum dots, Physical Review B 97 (2018) 085421.

Figure 1: Target structure and multi-scale scheme of device modeling. (a) A 3D
view of the silicon (Si) double quantum dot (DQD) structure that resembles the
physical one reported by Zajac $et$ $al.$ (Ref. [10]). Here the lateral
confinement (along the [100] direction) is controlled with DC biases imposed
on the top electrodes, while the vertical one (along the [010] direction) is
naturally formed due to the band offset among silicon-germanium (SiGe) and Si
layers. Since the structure is very long along the [001] direction, we use its
2D slice for device simulations assuming it is infinitely long along that
direction (a lateral distribution of the static magnetic field $B_{Z}$,
generated from a horseshoe-shaped cobalt micromagnet, is shown in the inset).
(b) The self-consistent loop of device simulations used to model spatial
distributions of charge and potential. Here the charge distribution at a given
potential distribution is obtained in two ways; the electronic structure
simulation based on a parabolic effective mass model is used to get the
density in Si layers where most of electrons reside, while the region of SiGe
layers is treated with the physics of bulk semiconductors.  Figure 2:
Incorporation of charge noise and steps for full-stack simulations covering
from device modeling to qubit operations. (a) The effect of charge noise is
incorporated into device simulations by disturbing the spatial distribution of
potential energy under no noise ($V_{IDEAL}$) with a random value that is
generated per real-space grid in a silicon (Si) double quantum dot (DQD)
structure under a zero-mean gaussian distribution of a standard deviation up
to 5$\mu$eV. (b) Starting with device modeling, our full-stack simulation can
eventually predict 2-qubit operations of a Si DQD system with the following
three steps: (1) device simulations that give bias-dependent energetic
positions and wavefunctions of electron spin states under no charge noise, (2)
disturbing the noise-free potential profile with charge noise, and (3) solving
a time-dependent Schrödinger equation for the Heisenberg Hamiltonian of two
neighbor spins that is constructed with results of device simulations.  Figure
3: Device initialization, 1-qubit rotation and single-step CNOT operation. (a)
A charge stability diagram of the double quantum dot (DQD) system that shows
electron-filling in each QD as a function of the left & right gate bias
($V_{L}$ & $V_{R}$). The middle gate bias ($V_{M}$) is set to 400mV. With
controls of gate biases, we can fill a single electron in each QD,
initializing the device to a $|$$\downarrow\downarrow$$\rangle$ state. (b)
Exchange ($J$) and Zeeman-splitting energy ($E_{ZL}$, $E_{ZR}$) of two QDs
shown as a function of $V_{M}$ when $V_{L}$ and $V_{R}$ are 540mV and 570mV,
respectively. Increasing $V_{M}$ lowers the potential barrier between two QDs
and enhances the interaction between electrons that occupy the down-spin
ground state of each QD. (c) Time responses of the DQD system at $V_{M}$ =
400mV. The $R_{Y}(\pi)$ operation for the left and right qubit is completed in
99.34 and 99.74 nanoseconds (nsec), respectively. When the interaction is
weak, we can address each qubit independently by setting the frequency of an
AC microwave pulse equal to the ground state Zeeman-splitting energy of each
QD, which is 18.31GHz (left) and 18.45GHz (right) in our case. (d) Time
responses simulated at $V_{M}$ = 408mV that achieve a single-step completion
of the controlled-NOT operation in 100.4nsec. A 8mV increase of $V_{M}$
dramatically enhances the interaction of QDs so $J$ at $V_{M}$ = 408mV is
$\sim$250 times larger than the case of $V_{M}$ = 400mV. Once QDs strongly
interact, the Rabi frequency of the qubit in one QD depends on the state of
the qubit in the other QD, generating 2-qubit entanglement.  Figure 4: Noise-
driven behaviors of 1-qubit rotation and single-step CNOT operation. (a) The
fidelity of a $R_{Y}(\pi)$ operation conducted with the left and the right
quantum dot (QD) are presented as a function of the magnitude of charge noise
($\sigma$: standard deviation of noisy potential values that are randomly
generated per grid of the modeling domain). The fidelity obtained with 1,000
samples is 99.93$\pm$10-6% (left QD) and 99.98$\pm$10-6% (right QD) when
$\sigma$ = 10${}^{-3}\mu$eV, and becomes 96.95$\pm$0.5664% and
96.98$\pm$0.5687% when $\sigma$ = 5$\mu$eV. (b) The single-step controlled-NOT
operation turns out to be much more vulnerable to charge noise than the case
of 1-qubit rotations so the fidelity becomes 98.34$\pm$0.003% and
32.84$\pm$0.5361% when $\sigma$ is 10${}^{-3}\mu$eV and 5$\mu$eV,
respectively. (c) Noise-driven fluctuation in Zeeman-splitting energy
($E_{ZL}$, $E_{ZR}$) and (d) exchange interaction ($J$) is shown in the regime
of weak ($V_{M}$ = 400mV) and strong interaction ($V_{M}$ = 408mV), revealing
that the degradation in fidelity, particularly in the case of entangling
operation, is due to the sensitivity of $J$ to charge noise.  Figure 5: Multi-
step CNOT operation with DC entangling logic. (a) (i) The CNOT operation can
be achieved with three steps, where the second one (a controlled-Z (CZ) gate)
consists of a 2-qubit unitary $U$ and two 1-qubit rotation blocks
($R_{Z}(-\pi/2)$) that can be implemented with only DC biases in double
quantum dot (DQD) platforms. Here the entanglement is generated by $U$. (ii) A
real-time control of the middle gate bias ($V_{M}$) for the three-step CNOT
operation when the left and right gate bias are 540mV and 570mV, respectively.
A bias-transition time ($\tau_{TR}$) of 5 nanoseconds (nsec) is assumed. (iii)
Resulting 2-qubit time responses show that the CNOT operation is completed at
132.1nsec. (b) The fidelity of a $U$ and a CZ block are shown as a function of
the magnitude of charge noise, indicating that 1-qubit Z-rotations do not
quite affect the preciseness of the overall CZ operation. (c) The multi-step
CNOT operation is much more robust to charge noise than the single-step case,
mainly due to the noise-robustness of the DC entangling block $U$.  Figure 6:
Acceleration of multi-step CNOT operation with $V_{M}$ control. (a) 2-qubit
time responses are simulated for the three cases, where the middle gate bias
($V_{M}$) is set to 408mV, 410mV, and 412mV to place the device in a regime of
strong interaction. The reference case (408mV) takes 25.9 nanoseconds (nsec)
to finish controlled-phase operation ($U$), and this time consumption becomes
7.2nsec and 1.9nsec when $V_{M}$ is 410mV and 412mV, respectively. (b)
Corresponding fidelities are plotted as a function of the magnitude of charge
noise, where we find the noise-driven degradation in fidelity of the second
case (410mV) does not show remarkable difference compared to the reference
case though 18.7 nsec can be saved for gating. The last case (412mV) is more
robust to charge noises, but the average fidelity is not good even at $\sigma$
= 10-3$\mu$eV (75.1%) due to the transition of $V_{M}$ ($\tau_{TR}$ = 5nsec),
as shown in (c) the loss in fidelity that is calculated as a function of
$\tau_{TR}$.