requires higher level of fine tuning and device uniformity, and for this
reason, in our modeling we have focused on the CB mode, for which these
limitations are less severe. We have predicted a relatively wide range of
parameters needed for stable operation of the CB SQuS, and proved its
robustness against simulated electrostatic disorder of realistic amplitude,
see Tab. 1 and Sec. III for discussion.

We have discussed two mechanisms of spin dephasing of the spin qubit shuttled
in the CB mode: one due to position-dependent spin precession frequency during
adiabatic shuttling, and another due to level-dependent spin precession
frequency. The latter mechanism is activated by non-adiabatic transitions (see
Sec. V for general discussion) between the orbital and valley degrees of
freedom, analyzed in detail in Sec. VI and Sec. VII, respectively. The
dephasing due to the first mechanism is suppressed by increasing shuttling
velocity $v$, which decreases the time the electron spends in the SQuS, and
activates the motional narrowing effect. The second mechanism becomes more
dangerous with increasing $v$ due to larger probability of the electron
straying away from adiabatic evolution trajectory.

Figure 15: Expected loss of spin coherence $\delta C$ experienced by an
electron shuttled through $10$-$\mu$m long SQuS as a function of electron
velocity $v$ (lower axes) and shuttling time $\tau$ (upper axes). We compare
various contribution to such an error: spatial variation of precession
frequency (red lines, Sec. IV), single spin-valley hotspot due to intrinsic
spin-orbit coupling (yellow, Sec. VII.5) and non-adiabatic effects in valley
degrees of freedom caused by: smooth gradient (green, Sec. VII.2) and sharp
atomistic steps (blue, Sec. VII.3). We distinguish between the magnetic noise
due to nuclear spins (darker red line) and the charge noise (lighter red
line). We highlight that hotspot contribution is the only one caused by spin
relaxation process, for which $\delta p_{\uparrow}\sim\delta C$. The
corresponding error should be multiplied by number of hotspots in the SQuS
(i.e. no hotspots for $E_{Z}\ll E_{\text{VS}}$). (a) $\delta C$ for relevant
values of effective valley splitting
$E_{\text{VS},0}=100,200,500\,\mu\mathrm{eV}$ (dashed, dashed-dotted and solid
lines respectively) for fixed $E_{\text{orb}}=2\,$meV and
$T_{2}^{*}=20\,\mu$s. (b) $\delta C$, but for different values of orbital
energy $E_{\text{orb}}=1,2,3$ meV (dashed, dashed-dotted and solid lines
respectively) for fixed values of $E_{\text{VS},0}=500\,\mathrm{\mu eV}$ and
$T_{2}^{*}=20$ $\mu$s. The results are drawn for the gradient of $a_{x}\approx
0.05\pi/$ nm (corresponding to intentional miscut angle $\vartheta\approx
0.5^{\circ}{}$), average step separation $\overline{d}=30$ nm
($\vartheta\approx 0.25^{\circ}$) and magnetic field of $B=100$ mT. The effect
of decreasing magnetic field to $B=20$ mT is illustrated by a shaded regions.
For plotted range of velocities $v\ll L_{\text{dot}}E_{\text{orb}}/\hbar$,
$\delta C$ due to orbital non-adiabaticity is exponentially small. For
comparison in a) we use black, dotted line to plot probability of spin-
relaxation during the transfer from [33] and Eq. (14)

The summary of the results of the last four Sections is given in Fig. 15,
where we focus on spin coherence of the qubit transferred using the CB mode.
In this Figure, the phase error, $\delta C$, is plotted against the electron
velocity $v$ for the relevant ranges of remaining parameters: typical
coherence times $T_{2}^{*}$ in stationary QD, ranges of orbital splitting
$E_{\text{orb}}$, and the bare valley splitting $E_{\text{VS},0}$. The last
parameter corresponds to typical value of valley splitting averaged over alloy
disorder, in absence of additional source of interface inhomogeneity (either
atomistic steps or a smooth gradient of interface position that we have
considered here). It means that the $E_{\text{VS},0}$ is an upper bound for
$E_{\text{VS}}$ that could be measured in a QD localized at some position.

The most important qualitative result shown in Fig. 15, is the prediction of a
relatively wide velocity range for the operation of the SQuS with coherence
error $\delta C$ below the targeted threshold of $10^{-3}$. The presence of
optimal $v$, which leads to smallest $\delta C$, results from existence of the
two above-mentioned dephasing mechanisms with opposite dependence of resulting
error on $v$. This range extends over at least one order of magnitude of
velocities, and it includes $v\\!=\\!10\,\mathrm{m/s}$ value, for almost all
combinations of parameters. Let us recall that simulations of the QD velocity
from Fig. 7f have shown that for a nominal $v$ of 10 m/s the velocity might
randomly increase up to $\approx\\!20$ m/s. Such a range of $v$ does not
endanger the targeted $\delta C\\!<\\!10^{-3}$, if the valley splitting is
sufficiently large, i.e. $E_{\text{VS,0}}>200\,\mu$eV.

The velocity of $10\,\mathrm{m/s}$ lies close to the boundary below which the
shuttling time becomes larger than typical qubit manipulation time (let us
recall that for shuttling distance of $L_{\mathrm{s}}=10\,\mu$s, shuttling
time of $\tau\\!\approx\\!1\,\mu\mathrm{s}$ requires $v\approx
10\,\mathrm{m/s}$). If slower shuttling could be accepted, larger values
$T_{2}^{*}$ would allow for $\delta C\\!<\\!10^{-3}$ for lower $v$: in Fig.
15, we see that for the $T_{2}^{*}\\!=\\!20\,\mu\mathrm{s}$ the phase error
due to quasistatic spin splitting noise grows above the $10^{-3}$ threshold
only for velocities below $1\,\mathrm{m/s}$. Another lower bound on the SQuS
electron velocity is the spin relaxation at the spin-valley hotspots. A single
hotspot leads to phase error indicated by the yellow line in Fig. 15 in the
absence of a strong magnetic field gradient. This error is the only one
considered here that is resulting from the spin relaxation process, as the
spin-orbit induced relaxation caused by electron motion has much weaker effect
[33].

On the other hand, the upper bound on $v$ is given by the dephasing caused by
intervalley excitations. According to Fig. 15, it is given by
$v_{\text{max}}\\!\approx\\!10\,\mathrm{m/s}$, in the most pessimistic
scenario of $E_{\text{VS},0}\\!=\\!100\,\mu\mathrm{eV}$, and can be as large
as $v_{\text{max}}\approx 50$ m/s for
$E_{\text{VS},0}\\!=\\!500\,\mu\mathrm{eV}$. In order to distinguish between
the cases relevant for gradient or steps models, we have considereded various
average tilts of the quantum well interface, which result in the gradient of
$a_{x}^{\text{max}}\\!=\\!\pi/L$ (corresponding to miscut angle
$\vartheta\\!=\\!0.5^{\circ}{}$) and the step separation
$\overline{d}\\!=\\!30\,\mathrm{nm}$ ($\vartheta\\!=\\!0.25^{\circ}{}$). With
those values, both models reproduce reported values of local valley splittings
$E_{\text{VS}}\\!\approx\\!(0.1-0.2)E_{\text{VS,0}}\in(10-100)\mu$eV. The
upper bound on the electron velocity $v_{\text{max}}$ will significantly
increase for smaller gradients and lower density of steps. In contrast,
lateral distance $d$ among the atomic steps being smaller than the QD size
$\overline{d}<L$ may result in unfavourable collapse of the valley splitting
and strongly nonadiabatic evolution, see Sec. VII for details.

We have considered charged defects at the oxide interface as the dominant
source of potential disorder in Si/SiGe. The impact of other crystalline
imperfections of the Si/SiGe heterostructure e.g. charged defects close to the
QW and the SQuS channel, or threading dislocations, may be relevant and need
to be investigated in the future. The more localized imperfections might more
easily lead to the orbital excitation of the shuttled electron. However, we
have shown that even extremely large orbital excitation rates, as high as
$100\,\mathrm{MHz}$, are unproblematic due to the fast orbital relaxation
rate.

From our analysis it is clear that operating at small magnetic field is
beneficial for coherent transfer. It minimizes both the phase error during
non-adiabatic transitions, and the probability of the occurrence of a spin-
valley hotspot at which $E_{Z}\\!=\\!E_{\text{VS}}$. In Sec. VII.5, we have
shown that the presence of the hotspot can be avoided with magnetic field
below $B\leqslant 100\,\mathrm{mT}$ ($E_{z}\leqslant 12\,\mu\mathrm{eV}$),
provided the valley splitting along the SQuS is relatively stable, i.e.
$E_{\text{VS}}>20\,\mu$eV. Otherwise, presence of the hotspots along the SQuS
introduces shuttling error due to spin relaxation. We have shown that for the
error threshold of $10^{-3}$, a typical error at the hotspot due to intrinsic
spin-orbit mechanism allows for crossing a few of them. In contrast, strong
magnetic field gradients (artificial/synthetic SOI) used for fast EDSR qubit
manipulation, may introduce spin-flip error above the threshold even at a
single hotspot. Such large gradients are not required in a shuttling based
architecture, since qubits can be transported to dedicated manipulation zones,
in which the displacement amplitude of the electron is larger during EDSR.
Hence, orders of magnitude smaller gradients may be sufficient to reach the
same EDSR Rabi frequency. Notably, in the absence of hotspot occurrences, the
remaining discussed relevant error processes discussed at length in the paper,
are of pure dephasing character. This may be utilized in tailoring quantum
error correction schemes for architectures using SQuS devices as coherent
links.

We have found that for multiple reasons, increasing the effective valley
splitting $E_{\text{VS,0}}$ strongly improves operation of the SQuS. Firstly,
the number of spin-valley hotspots discussed above can be reduced, if the
valley splitting is sufficiently large and uniform along the SQuS, i.e.
$E_{\text{VS}}\gg E_{z}$. Secondly, large $E_{\text{VS,0}}$ significantly
limits dephasing caused by valley non-adiabaticity: on one hand it limits
excitation probability in both considered interface models (Fig. 15), and on
the other hand it enhances valley relaxation via phonon emission (Fig 13).
Recently, a few methods of increasing valley splitting $E_{\text{VS,0}}$ in
stationary QDs have been investigated. One of them suggests back-gates for
controlling the electric field across the QD independent from the QD filling
[114]. Another approach uses engineering of the Ge profile across the Si/SiGe
heterostructure [79, 115]. A third method relies on increasing random
fluctuations of alloy composition (Ge concentration in Si QW), which
statistically increases average $E_{\text{VS,0}}$ [78], but at the cost of
larger variance. Thus, from the perspective of the SQuS, it might increase the
probability of unfavourable regions with small valley splitting, and also
increase spatial variation of valley field, which could be translated to
larger gradients and less regular steps.

The results presented here show that even for the smallest considered
$E_{\mathrm{VS,0}}\\!=\\!100\,\mu\mathrm{eV}$ it should be possible to
transfer the electron spin qubit over $10\,\mu\mathrm{m}$ in about
$1\,\mu\mathrm{s}$ with phase error of $10^{-3}$. However, having a wide range
of parameters allowing for such coherent transfer is clearly desirable. Let us
now discuss a few methods to make the operation of the SQuS become more robust
and faster.

The effects of the electric noise can be diminished to some extent by
populating the SQuS with multiple electrons, creating an effective screening
effect [116, 117]. For example, filling the shuttled QD with three electrons,
the ground valley states are fully occupied by a spin singlet state. In the
opposite regime of low velocities, the noise might be dominated by the
magnetic noise either from nuclear spins, or spatial variation of g-factor. In
such case, a similar effect can be achieved by shuttling the spin qubit in the
singlet/unpolarized triplet basis, which in principle should be less sensitive
to correlated noise, and requires filling the moving quantum QD with two
electrons.

Naturally, there are numbers of applicable methods if the SQuS is relatively
short, and thus a certain level of active tuning is feasible: The limitation
of compensating potential disorder imposed by our gate sets approach can be
circumvented by tuning the voltages of a SQuS in the time domain. This
strategy assumes that only one qubit is shuttled in the SQuS at a time. Since
its position is deterministic, disorder induced potential offsets can be
compensated by tuning the voltage signals applied to the SQuS in real time.
Also, the electron velocity can be adapted to disorder/noise in certain
regions of the SQuS. In particular, the possibility of individual tuning of
the gates could make the coupling between the predefined QDs more uniform,
which in principle could enhance the feasibility of the bucket-brigade mode
shuttling. Besides the larger effort of tuning, this approach requires much
higher complexity of the control signals, the generation of which is more
involved and particularly limited by the space for memory of on-chip control
electronics.

Let us finish with stressing that the presented analysis of sources of errors
caused by shuttling of electron in Si/SiGe structures highlights two areas for
future research. As the main decoherence mechanism at higher shuttling
velocities is due to valley excitations caused by atomic disorder at Si/SiGe
interface in presence of valley-dependence of electron spin $g$-factors,
further research into interface disorder and $g$-factor physics in Si/SiGe
structures is needed. With new insights on the influence of material growth
and nanostructuring on atomic disorder at the interface, strategies to
decrease both the probability of valley excitations, and the valley-dependence
of $g$-factors, may emerge.

Finally, we note that a similar architecture can be used in other
semiconductor-based quantum devices, with the relevant example of GaAs-based
nanostructures, in which the absence of valley degree of freedom may allow for
faster operation of the SQuS that reduces the detrimental influence of the
nuclear spins. Other systems, such as hole spins in Ge/SiGe [118] avoid
challenges of both coupling to nuclear spins and relevance of the valley
degeneracy, but the stronger spin-orbit interaction of hole qubits need to be
considered.

##  IX Acknowledgements

We thank Hendrik Bluhm and Lieven Vandersypen for valuable discussions and
David DiVincenzo for comments on the manuscript. This work has been funded by
the National Science Centre (NCN), Poland under QuantERA program, Grant No.
2017/25/Z/ST3/03044, and under the ETIUDA doctoral scholarship, Grant No.
2020/36/T/ST3/00569 (J.A.K.), and by the Federal Ministry of Education and
Research under Contract No. FKZ: 13N14778 as well as by the German Research
Foundation (DFG) within the projects BO 3140/4-1, and under Germany’s
Excellence Strategy - Cluster of Excellence Matter and Light for Quantum
Computing" (ML4Q) EXC 2004/1 - 390534769289786932. Project Si-QuBus has
received funding from the QuantERA ERA-NET Cofund in Quantum Technologies
implemented within the European Union’s Horizon 2020 program.

V.L. and J.A.K. contributed equally to all the theory presented in this work.
N.F. did numerical device simulations and I.S. fabricated the described
device. L.R.S. and Ł.C. equally supervised the work. All authors extensively
discussed all sections presented in this work and jointly wrote the
manuscript.

##  Appendix A Details on geometry estimates for the QuBus gate design

Here, we provide a Fourier mode analysis to arrive at the scaling relations
presented in Sec. III.1. Choosing the lowest non-trivial mode of the
confinement potential along the $x$-direction (a sine wave with angular wave
number $K_{x,1}=2\pi/L_{x}$) as the relevant one, the curvature of the QD
potential $c_{\mathrm{conf}}$ will simply be proportional to $K_{x,1}^{2}$.
However, the overall amplitude of the mode is given by the mode decay when
propagating the mode from the clavier gate region down into the quantum well.
We solve a 2D Laplace equation to account for the propagation of the amplitude
of the Fourier modes:

$\left(\frac{\partial^{2}}{\partial z^{2}}-K_{x,n}^{2}\right)c_{n}(z)=0$ (38)

For a homogeneous medium, the mode amplitude then straightforwardly obeys an
exponential decay with the corresponding angular wave number $K_{x,n}\equiv
2\pi n/L_{x}$:

$c_{n}(-h)=c_{n}(0)e^{-K_{x,n}h},$ (39)

where $h=h_{\mathrm{SiGe}}+h_{\textbf{}}{ox}$ is the distance of the
considered QW-plane to the lower face of the clavier gates and $c_{n}$ are the
Fourier coefficients. We may then introduce a confinement coefficient by
perform a harmonic approximation at the minimum of the first harmonic $n=1$,
which then scales as

$c_{\mathrm{conf}}\propto\frac{1}{L_{x}^{2}}c_{1}(z=-h_{\mathrm{SiGe}}).$

Assuming the depth of the QW to be fixed and varying only the conveyor unit
cell length $L_{x}$, the maximum of this confinement coefficient then occurs
at $L_{x}=L_{\mathrm{opt}}\equiv\pi(h_{\mathrm{SiGe}}+h_{\mathrm{ox}})$ in the
case of homogeneous dielectrics. Note that the possible depth of the QW is
constrained by the electrostatic requirements for qubit manipulation thus the
distance to the charged defect is limited. Choosing this optimal SQuS cell
length, the confinement coefficient scales as

$c_{\mathrm{conf}}\propto\frac{1}{L_{\mathrm{opt}}^{2}}c_{1}(z=-h_{\mathrm{SiGe}})\big{|}_{L_{x}=L_{\mathrm{opt}}}.$

With
$c_{1}(z=-h_{\mathrm{SiGe}})\big{|}_{L_{x}=L_{\mathrm{opt}}}=c_{n}(0)e^{-2}=\mathrm{const}$,
only the inverse quadratic scaling
$c_{\mathrm{conf}}\propto\frac{1}{L_{\mathrm{opt}}^{2}}$ remains, and with
$E_{\mathrm{orb}}\propto\sqrt{c_{\mathrm{conf}}}$ in a harmonic potential
approximation, $E_{\mathrm{orb}}\propto\frac{1}{L_{\mathrm{opt}}}$ will be
approximately inversely proportional to the depth of the quantum well.

To be more precise, one may take the different dielectric constants in the
oxide (3.9 for SiO2) and semiconductor regions (13.2 averaged over alloy
composition for Si0.7Ge0.3) into account. For that, we need to choose a
slightly more involved exponential ansatz and work the dielectric constants
into the constraint of continuity of electric flux density at the boundary
between the two media. For convenience, we choose $z=0$ as the position of the
semiconductor-oxide interface. This yields

$c_{n}(z=-h_{\mathrm{SiGe}})=c_{n}(h_{\mathrm{ox}})\frac{e^{-K_{x,n}h_{\mathrm{SiGe}}}(1+r)}{e^{+K_{x,n}h_{\mathrm{ox}}}+re^{-K_{x,n}h_{\mathrm{ox}}}}$
(40)

with
$r\equiv(\varepsilon_{r,\mathrm{ox}}-\varepsilon_{r,\mathrm{SiGe}})/(\varepsilon_{r,\mathrm{ox}}+\varepsilon_{r,\mathrm{SiGe}})$.
Now we discuss the maximization of the confinement coefficient

$c_{\mathrm{conf}}\propto\frac{1}{L_{x}^{2}}c_{1}(z=-h_{\mathrm{SiGe}}).$

We will first assume the depth of the QW to be fixed and vary only the
conveyor unit cell length $L_{x}$. The maximum then occurs at
$L_{\mathrm{opt,0}}=\pi(h_{\mathrm{SiGe}}+h_{\mathrm{ox}})$ for the
homogeneous case. Taking the different dielectric constants of the oxide and
semiconductor into account, we have to solve the nonlinear equation

$L^{\mathrm{inh}}_{\mathrm{opt}}=\pi(h_{\mathrm{SiGe}}+h_{\mathrm{ox}}f(2\pi/L^{\mathrm{inh}}_{\mathrm{opt}}))$
(41)

with

$f(K_{x})=\frac{e^{+K_{x}h_{\mathrm{ox}}}-re^{-K_{x}h_{\mathrm{ox}}}}{e^{+K_{x}h_{\mathrm{ox}}}+re^{-K_{x}h_{\mathrm{ox}}}}.$

Using the homogeneous solution as the argument for the right hand side of
(41), already gives a very good approximation of the optimal period length

$L^{\mathrm{inh}}_{\mathrm{opt}}\approx\pi(h_{\mathrm{SiGe}}+h_{\mathrm{ox}}f(2\pi/L_{\mathrm{opt}}))$
(42)

for our discussion.

Figure 16: Orbital-splitting as a function of the geometric parameters of the
SQuS without disorder. (a) Minimum orbital splitting $E_{\mathrm{orb}}^{x}$
extracted from the first mode of the Fourier analysis as a function of the
SQuS unit cell length $L_{x}$ and the thickness of the SiGe layer
($h_{\mathrm{SiGe}}$). (b) Calculation of the minimum orbital splitting
$E_{orb}^{x}$ from solving the Schrödinger equation on the full-mode periodic
potential. In panels a to b, $h_{ox}=10$ nm and $V_{a}=100$ mV is fixed and
the red line indicates $L_{\mathrm{opt}}^{\mathrm{inh}}$ and the red dashed
line $L_{\mathrm{opt}}$.

The behavior of the orbital confinement is plotted in Fig. 16, illustrating
the degree of deviation of $L_{\mathrm{opt}}$ from the inhomogeneous estimate
$L_{\mathrm{opt}}^{\mathrm{inh}}$ and the appropriateness of the first
harmonic approximation Fig. 16a when compared to the exact solution in Fig.
16b. We note that $L^{\mathrm{inh}}_{\mathrm{opt}}$ scales approximately
linearly with the QW depth parametrized by $h_{\mathrm{SiGe}}$, mostly
shifting the solution for the homogeneous case (cf. Fig. 6a and 6b).

##  Appendix B Formal treatment of spin dephasing due to orbital/valley
transitions

Here we extend the analysis of electron dephasing due to relaxation from an
excited to the ground orbital/valley state, which has been effectively
introduced in Sec. V. For simplicity, we assume that we are dealing with two
adiabatic orbital/valley states, $\ket{g}$ and $\ket{e}$, each with a constant
$g$-factor. We assume that at an initial time $t_{e}$ an event leading to
finite population of $\ket{e}$ state happens. Such an event is associated with
a momentary decrease of the energy gap between these two states, and/or
increase of motion-induced coupling between them. We look at the subsequent
evolution of the qubit, while taking the temperature of the environment as
much smaller than the post-excitation event energy difference between
$\ket{e}$ and $\ket{g}$ states, so that the environment can only cause
transitions from $\ket{e}$ to $\ket{g}$ state.

We describe the electron spin dynamics in a reference frame rotating with
Larmor precession of spin of electron in $\ket{g}$ state, so that there is no
spin dynamics in this state, and the spin of electron in $\ket{e}$ is rotating
about $z$ axis according to the Zeeman Hamiltonian given by

$\hat{H}_{Z}=\ket{e}\bra{e}\otimes\hbar\delta\omega\hat{\sigma}_{z}/2\,\,,$
(43)

where $\hbar\delta\omega=2(\delta g/g)\mu_{B}B$. We assume that the electron
had been prepared in the past in $\ket{g}$ state, with its spin in a
superposition state
$\ket{\Psi_{s}}=a_{\uparrow}\ket{\uparrow}+a_{\downarrow}\ket{\downarrow}$, so
that the initial coherence is
$C(-\infty)\\!=\\!a_{\uparrow}a^{*}_{\downarrow}$. The process of excitation
at time $t_{e}$ is treated for simplicity as point-like in time. It can be
coherent, due to Landau-Zener transition that is not fully adiabatic, leading
to a creation of superposition of electron being in ground $\ket{g}$ and
excited $\ket{e}$ states. It can also be incoherent, due to charge disorder
acting as noise in reference frame co-moving with the QD, leading only to
creation of occupation of higher-energy state $\ket{e}$. In the first case,
the electron at time $t_{e}$ is in a pure state,

$\ket{\Phi(t_{e})}=(a_{g}\ket{g}+a_{e}\ket{e})\otimes(a_{\uparrow}\ket{\uparrow}+a_{\downarrow}\ket{\downarrow})\,\,,$

and the corresponding density operator $\ket{\Phi(t_{e})}\bra{\Phi(t_{e})}$
has all its matrix elements finite (assuming none of $a_{i}$ amplitudes is
zero). In the second case the density operator corresponding to the created
partially excited state is

$\hat{\rho}(t_{e})=\big{[}(1-p_{e})\ket{g}\bra{g}+p_{e}\ket{e}\bra{e}\big{]}\otimes\ket{\Psi_{s}(t_{e})}\bra{\Psi_{s}(t_{e})}\,\,,$
(44)

which has zero $e$-$g$ coherences, i.e.
$\bra{e}\hat{\rho}(t_{e})\ket{g}\\!=\\!0$. In both cases the spin coherence in
$\ket{g}$ state is diminished compared to pre-excitation value, as it is
multiplied by $|a_{g}|^{2}$ or $1-p_{e}$ factors, respectively. The “missing”
part of spin coherence is in the $\ket{e}$ state, and we want to see if
subsequent evolution, involving energy relaxation into the environment, can
bring back this missing part while returning the electron into the adiabatic
ground state $\ket{g}$.

The evolution for $t\\!>\\!t_{e}$ is then due to $\hat{H}_{Z}$ from Eq. (43)
and to transitions from $\ket{e}$ to $\ket{g}$ state caused by interaction
with the environment. The first process is unitary, and for
$\delta\omega\\!\neq\\!0$ it correlates the spin degrees of freedom with $e/g$
orbital or valley degrees of freedom. The second process in nonunitary, and we
describe it using the Born-Markov approximation (which is definitely
appropriate for treating phonon-induced orbital and valley relaxation in
quantum dots), which leads to Lindblad equation for evolution of density
matrix:

$\displaystyle\frac{\mathrm{d}\hat{\rho}(t)}{\mathrm{d}t}$
$\displaystyle=-i[\hat{H}_{Z},\hat{\rho}(t)]+$
$\displaystyle\sum_{k=1}^{n}\left[\hat{L}_{k}\hat{\rho}(t)\hat{L}^{\dagger}_{k}-\frac{1}{2}\hat{L}^{\dagger}_{k}\hat{L}_{k}\hat{\rho}(t)-\frac{1}{2}\hat{\rho}(t)\hat{L}^{\dagger}_{k}\hat{L}_{k}\right]\,\,,$
(45)

where $n$ is the number of jump operators $L_{k}$.

We neglect spin-orbit coupling in treatment of electron-phonon interaction, so
that we disregard phonon-induced spin relaxation (its significance has been
discussed in Sec. VII.5). The $\hat{L}_{k}$ operators must then be spin-
diagonal. Their dependence on spin degree of freedom is not however obvious:
for $\delta\omega\\!\neq\\!0$ the transitions for the two spin directions
correspond to transfer of unequal energy quanta into the environment.

If time-energy uncertainty allows for resolving of the $\delta\omega$ energy
difference, which is the case when $\delta\omega\\!\gg\\!\gamma$, where
$\gamma$ is the $e$-$g$ transition rate, we should use a separate jump
operator $\hat{L}_{k}$ for each spin. For phonon emission, the physical
picture is the following: in this situation the phonon wave packets emitted
for transitions involving each spin are not overlapping in frequency, so in
principle the information on which spin-diagonal transition occurred is
imprinted on the environment [119], and the act of phonon emission amounts to
measurement of the spin projection. It is straightforward to check that if we
use $\hat{L}_{s}=\sqrt{\gamma}_{s}\ket{g}\bra{e}\otimes\ket{s}\bra{s}$ with
$s\\!=\\!\uparrow$, $\downarrow$, there is no pumping of coherence from
$\rho_{e\uparrow,e\downarrow}$ to $\rho_{g\uparrow,g\downarrow}$, and while
the relaxation leads to repopulation of $\ket{g}$ state, i.e. $\rho_{gs,gs}$
occupation returns to its pre-excitation value, spin coherence remains
suppressed by $1-p_{e}$ factor. Note that using the terminology of Section V,
$\delta\omega\\!\gg\\!\gamma$ is equivalent to
$\delta\phi\\!\approx\\!\delta\omega/\gamma\\!\gg\\!1$, so that we recover the
result of $\delta C\\!\approx\\!p_{e}$.

On the other hand, when $\delta\omega\\!\ll\\!\gamma$, the traces left in the
environment by transitions involving each of the spin states are
indistinguishable, and we should use a single jump operator that is blind to
the spin degree of freedom,
$\hat{L}_{0}\\!=\\!\sqrt{\gamma}\ket{g}\bra{e}\otimes\mathds{1}_{s}$ where
$\mathds{1}_{s}$ is an unit operator in spin space. With such a jump operator,
Eq. (45) leads to the following time evolution of the ground-state spin
coherence

$\rho_{g\uparrow,g\downarrow}(\Delta
t)=\rho_{g\uparrow,g\downarrow}(t_{e})+\rho_{e\uparrow,e\downarrow}(t_{e})\frac{\gamma\left(1-e^{-i\delta\omega\Delta
t}e^{-\gamma\Delta t}\right)}{\gamma+i\delta\omega}\,\,,$ (46)

in which $\Delta t\\!=\\!t-t_{e}$. For $\gamma\Delta t\\!\gg\\!1$ we obtain
the asymptotic value of coherence after relaxation:

$|\rho_{g\uparrow,g\downarrow}(\Delta t\gg
1/\gamma)|\approx|C(-\infty)|\left|1-p_{e}+p_{e}\frac{\gamma}{\gamma+i\delta\omega}\right|\,\,,$
(47)

in which we used
$\rho_{g\uparrow,g\downarrow}(t_{e})\\!=\\!(1-p_{e})C(-\infty)$ and
$\rho_{e\uparrow,e\downarrow}(t_{e})\\!=\\!p_{e}C(-\infty)$. Expanding this
result to the lowest order in $p_{e}$ and $\delta\omega/\gamma\\!\ll\\!1$ we
obtain $\delta C\approx p_{e}\delta\omega^{2}/\gamma^{2}$, which is in
agreement with the previous qualitative estimate $\delta
C\\!\approx\\!p_{e}\delta\phi^{2}/2$ valid for $\delta\phi\\!\ll\\!1$ once we
identify $\delta\omega/\gamma$ (equal to $2(\delta g/g)\mu_{B}B\tau_{e}$ using
the quantities used in Section V) with $\delta\phi/\sqrt{2}$.

##  Appendix C Effects of the electrostatic disorder on the electron in the
moving dot

Let us first reconstruct the correlation function of electrostatic disorder
felt by the electron in the ground state of a moving dot. Following definition
from Eq. (VI.1) we write such a correlation function as:

$\displaystyle K_{\text{gg}}(\Delta x)=\int\text{d}x_{1}\text{d}x_{2}$
$\displaystyle|\psi_{g}(x_{1})|^{2}|\psi_{g}(x_{2})|^{2}$ $\displaystyle
K_{\delta V}(x_{1}-x_{2}+\Delta x).$ (48)

In the above the bare correlation function of electrostatic disorder is
assumed to be translationally invariant, and have an exponential form:

$K_{\delta V}(x)=\langle\delta V(x)\delta V(0)\rangle=\delta
V^{2}e^{-|x|/l_{c}^{\delta V}},$ (49)

which is parameterized by the amplitude $\delta V$ and correlation length
$l_{c}^{\delta V}$. Using Fourier transform, the $K_{\text{gg}}(\Delta x)$ can
be written as:

$K_{\text{gg}}(\Delta x)=\int\frac{\text{d}k}{2\pi}\frac{2\delta
Vl_{c}}{1+(kl_{c})^{2}}e^{ik\Delta x}|F_{\text{gg}}(k)|^{2},$ (50)

in which we introduced the Fourier transform of electron probability density:

$F_{\text{gg}}(k)=\int\text{d}x|\psi_{g}(x)|^{2}e^{-ikx}=e^{-\frac{1}{2}kL_{\text{dot}}^{2}}.$
(51)

In Fig. 17 we plot the numerically integrated Eq. (50), and compare against
the result of Fig. 9 in which $K_{\text{gg}}(x)$ was obtained by averaging
over realisations of the electrostatic disorder. In this way prove that the
exponential form of $K_{\delta V}(x)$ allows for reconstruction of the
numerical result.

Figure 17: Correlation function of the matrix element for the electron
occupying ground orbital state $K_{\text{gg}}(|x|)$. With dashed lines we plot
numerically integrated formulas Eq. (50) assuming exponential form of the
correlation function of effective 1D potential $\langle\delta V(x)\delta
V(0)\rangle=\delta V^{2}e^{-|x|/l_{c}^{\delta V}}$, with $l_{c}^{\delta
V}=90\,\mathrm{nm}$ and two values of noise amplitude $\delta
V=2.8\,\mathrm{meV}$ (blue) and $\delta V=3\,\mathrm{meV}$ (red) . For
comparison using green solid line we plot $K_{\text{gg}}(|x|)$, obtained from
the numerical averaging of different realisations of electrostatic disorder
(See Fig. 9).

.

We now use the exponential correlation function $K_{\delta V}(x)$ to compute
the transition rate from ground to excited orbital state. Following Eq. (11),
such a rate is related to the temporal Fourier transform of the correlation
function of the matrix element between ground and excited orbital states (see
Eq. (10)). In analogy to Eq. (50) such correlation function can be written as:

$K_{\text{eg}}(\Delta x)=\int\frac{\text{d}k}{2\pi}\frac{2\delta
Vl_{c}}{1+(kl_{c})^{2}}e^{ik\Delta x}|F_{\text{eg}}(k)|^{2}\,\,,$ (52)

in which the filtering of spatial disorder depends on the shape of the
wavefunctions of ground and excited orbital states. For harmonic potential we
have:

$F_{\text{eg}}(k)=\int\text{d}x\,\psi_{g}(x)\psi_{e}(x)e^{ikx}=ikL_{\text{dot}}e^{-\frac{1}{2}k^{2}L_{\text{dot}}^{2}}.$
(53)

For the moving dot the argument of $K_{\text{eg}}(\Delta x)$ becomes time-
dependent $\Delta x=vt$, which allows to compute the transition rate as:

$\displaystyle\Gamma_{+,o}$
$\displaystyle=\int\text{d}tK_{\text{eg}}(vt)e^{-iE_{\text{orb}}t}=$ (54)
$\displaystyle=\frac{\delta V^{2}l_{c}^{\delta
V}}{v}\frac{E_{\text{orb}}^{2}L_{\text{dot}}^{2}}{v^{2}+(E_{\text{orb}}l_{c}^{\delta
V})^{2}}\exp(-\frac{E_{\text{orb}}^{2}L_{\text{dot}}^{2}}{2v^{2}}),$

Finally in the experimentally relevant limit of $v\ll
E_{\text{orb}}\,l_{c}^{\delta V}$, we arrive at Eq. (12) from the main text.

##  Appendix D Landau-Zener approximation of multiple step passage

We provide here more detailed description of the electron moving over multiple
atomistic steps, the model of which has been used in Sec. VII.3. We derive
here the probability of coherent transition to excited valley state on a
single atomistic step, and then verify result of multiple step passage against
numerical simulations.

###  D.1 Single step passage

First we consider passage over single step localized at $x=0$. In the harmonic
approximation the probability of occupying left ($x<0$) and right regions
($x>0$) are given by $p_{\text{L}/\text{R}}(t)=\frac{1}{2}(1\mp\erf[vt/L])$.
For a step of single atomistic height, the relative valley phase between those
regions is shifted by $\Delta\varphi=0.85\pi$. For the symmetry reasons we
thus assume the valley phases are given by
$\varphi_{\text{VS},L}=-\Delta\varphi/2$ in the left and
$\varphi_{\text{VS},R}=\Delta\varphi/2$ in the right region. This allows to
write Hamiltonian (VII.1) in the form:

$H_{\text{v,s}}(t)=\frac{E_{\text{VS,0}}}{2}\bigg{(}\cos(\frac{\Delta\varphi}{2})\hat{\sigma}_{x}+\erf(vt/L)\sin(\tfrac{\Delta\varphi}{2})\hat{\sigma}_{y}\bigg{)}.$
(55)

Next we apply a simple basis transformation (rotation around x-axis), that
allows us to cast the above Hamiltonian into Landau-Zener form
$\hat{H}_{\text{LZ}}=\frac{1}{2}(\epsilon(t)\hat{\sigma}_{z}+\Delta\hat{\sigma}_{x})$,
i.e.:

$H_{\text{v,s}}(t)=\frac{E_{\text{VS,0}}}{2}\big{(}\cos(\tfrac{\Delta\varphi}{2})\hat{\sigma}_{x}+\erf(vt/L_{\text{dot}})\sin(\tfrac{\Delta\varphi}{2})\hat{\sigma}_{z}\big{)},$
(56)

where the $\erf(x)=(2/\sqrt{\pi})\int_{0}^{x}e^{-y^{2}}\mathrm{d}y$ is the
Gaussian error function. We can identify time-independent part as:

$\zeta=E_{\text{VS,0}}\cos(\tfrac{\Delta\varphi}{2}),$ (57)

while the time-dependent term reads:

$\epsilon(t)=E_{\text{VS,0}}\erf(vt/L_{\text{dot}})\sin(\tfrac{\Delta\varphi}{2}).$
(58)

Since L-Z formula of non-adiabatic exciation
$Q_{\text{LZ}}=\exp(-\pi\zeta^{2}/2\hbar a)$ is applicable for linear sweep,
$\epsilon(t)=at$, we linearize the error function around $t=0$, which produces
an effective sweep rate:

$a\approx\frac{2E_{\text{VS,0}}v}{\sqrt{\pi}L_{\text{dot}}}\sin(\tfrac{\Delta\varphi}{2}).$
(59)

This finally allow us to estimate probability of non-adiabatic excitation on a
single step as:

$\displaystyle Q_{1}$
$\displaystyle=\exp(-\frac{\pi}{2\hbar}\frac{\sqrt{\pi}L_{\text{dot}}}{2E_{\text{VS,0}}\sin(\tfrac{\Delta\varphi}{2})}E_{\text{VS,0}}^{2}\cos^{2}(\tfrac{\Delta\varphi}{2}))=$
$\displaystyle=\exp(-\pi^{3/2}\frac{E_{\text{VS},0}\,L_{\text{dot}}}{4\hbar
v}\frac{\cos^{2}\tfrac{\Delta\varphi}{2}}{\sin\tfrac{\Delta\varphi}{2}}).$
(60)

In Fig. 18 we plot occupation of excited valley state after single step
passage $Q_{1}$ as a function of $\eta=\hbar v/E_{\text{VS,0}}L_{\text{dot}}$.
In the inset we compare numerical solution (dots) against Landau-Zener
approximation (lines) for $\Delta\varphi/\pi=0.3,0.7,0.85,0.95$, which shows
that $Q_{1}$ formula is applicable in the relevant range of $\Delta\varphi\sim
0.85\pi$, i.e. for wide enough sweep of effective adiabatic parameter
$\epsilon(t)$. Otherwise, for small angles $\Delta\varphi$ the initial and
final adiabatic ground states would be not "orthogonal enough" since for small
angles $\epsilon(\infty)/t_{c}=\tan(\Delta\varphi/2)\approx\Delta\varphi/2$.

Figure 18: Occupation of higher valley state after single step passage $Q_{1}$
as a function of shuttling parameter $\eta$ (lower scale) and valley field
angle difference $\Delta\varphi$. Inset: Numerical solution to Eq. (22) is
compared against effective L-Z approximation (30) for selection of angles
$\Delta\varphi/\pi=0.3,0.7,0.85,0.95$ (dashed lines in larger figure). The
upper x-axis translates $\eta$ into electron velocity
$v=L_{\text{dot}}E_{\text{VS},0}\eta/\hbar$ assuming typical values of
$L_{\text{dot}}=20\,$nm and $E_{\text{VS},0}=200\,\mu$eV.

###  D.2 Numerical test of multiple step passage

In Sec. VII.3 we computed occupation of excited valley state by concatenating
number of isolated single-step passages. We argued that for small enough
probability of non-adiabatic excitation on a single step $Q_{1}\ll 1$, the
probability of coherent return to the ground state scales as $Q_{1}^{2}\ll
Q_{1}$ and hence can be neglected. As a result the probability of occupying
higher valley state is the effect of quantum interference between transitions
on single atomistic steps, i.e.

$p_{e,v}\approx Q_{1}\left|\sum_{n=1}^{N}e^{i\Phi_{n}}\right|^{2},$ (61)

which is bounded from above by $p_{e,v}\leq N^{2}Q_{1}$. Additionally we
argued that due to intrinsic random fluctuations of electron velocity the
possibility of coherent addition of probability amplitudes is negligibly
small, and in particular in the limit of large number of uncorrelated phases
gives $\big{|}\sum_{n=1}^{N}e^{i\Phi_{n}}\big{|}^{2}\sim N$.

Here we use Fig. 19 to test this hypothesis using numerical simulation of
multi-step passage, with quasistatic fluctuations in electron velocity. For
concreteness we take $v=30$ m/s and assumed its uncertainty (rms) to be
$\Delta v/v=10\%$. For each value of velocity we numerically solve the
evolution generated by the time-dependent Hamiltonian:

$\hat{H}_{\text{v,s}}(x_{0})=\frac{E_{\text{VS},0}}{2}\sum_{n=0}^{N}p_{n}(x_{0})\Big{[}\cos(n\Delta\varphi)\hat{\tau}_{x}+\sin(n\Delta\varphi)\hat{\tau}_{y}\Big{]},$
(62)

where $p_{n}(x_{0})=\int_{x_{n}}^{x_{n+1}}\rho(x-x_{0})\text{d}x$, in which
$\rho(x-x_{0})$ is the ground state probability density function of the
traveling electron centered at $x_{0}=vt$ and $x_{n}$ is the location of nth
step. By averaging over many realizations of electron velocity we obtain
average occupation of excited valley state:

$Q_{N}=\langle\Tr{\outerproduct{e_{v}}{e_{v}}\varrho(\tau)}\rangle_{v}.$ (63)

Figure 19: Probability of occupying higher valley state after N-step passage
$Q_{N}$. We use a simple model of disorder in which electron velocity $v$
fluctuates between the realizations (filled dots) with rms $\delta v=0.1v$,
and for remaining parameters we take $L=15$ nm, $E_{\text{VS},0}=100\,\mu$eV,
$\overline{v}=30$ m/s, $\vartheta=0.25^{\circ}{}$, which corresponds to
$\overline{d}\sim 30$ nm. To show statistical behaviour, for each number of
steps $N$ we draw a new arrangement of steps with terrace lengths given by the
distribution Eq. (64) and for that arrangement simulate 200 realisations of
velocity.

To reflect realistic conditions for fixed $N$ we use the same arrangement of
steps for all realization of electron velocity. However we change the
arrangement for each $N$ such that multiple arrangements are investigated. We
follow [105, 106] and draw length of consecutive terraces $d_{n}\equiv
x_{n+1}-x_{n}$ from independent Gaussian distributions:

$P(d_{n})\sim\mathcal{N}(\overline{d},(\overline{d}/5)^{2}),$ (64)

where $\overline{d}$ is the average distance between the steps, which can be
related to an average interface miscut $\theta_{\text{rad}}=h/\overline{d}$.
In Fig. 19 we plot the results of single realization (small dots) their
average (large hollow dots) and compare them against the coherent limit
$Q_{N}^{\text{max}}=N^{2}Q_{1}$ (red solid line) and average limit
$Q_{N}\approx NQ_{1}$ (green solid line) as a function of number of steps,
proportional to shuttling distance $L_{s}=N\overline{d}$.

As it can be observed the average occupation of excited valley state after
traveling over $N$ steps agrees with linear prediction, even for the
parameters which exaggerates typical valley excitation, i.e. $Q_{1}\approx
10^{-2}$ . What is more, single realizations of the shuttling are generally
contained below $N^{2}Q_{1}$ red line, with an exception of few points for
which a single realization of $v$ allowed for more then a single transition
between adiabatic levels. Notably most results lies also below the average,
the value of which is however increased by the outliers close to $N^{2}Q_{1}$
line. We stress out that the predictions are valid for typical for conveyor
$NQ_{1}\ll 1$, since otherwise one has to take into account more then a single
transition between the levels and more complex interference pattern.

##  Appendix E Spin-valley hot spot

###  E.1 Estimation of spin-valley coupling

We start the analysis by characterizing the spin-valley avoided crossing
visible in Fig. 14. To do so we estimate typical values of the spin-valley
coupling $\Delta_{\text{s-v}}$ (the gap) and the effective sweep rate
$a\equiv\mathrm{d}(E_{\text{VS}}(t)-E_{z})/\mathrm{d}t$, which together give
an estimate of time spent around the avoided crossing
$\tau_{\text{s-v}}\approx\Delta_{\text{s-v}}/a$. The spin-valley coupling can
be computed in second order perturbation theory as a result of spin-orbit and
valley-orbit couplings:

$\displaystyle\Delta_{s-v}^{\text{art}}\approx$ $\displaystyle 2\frac{|\langle
g_{o}\downarrow|H_{\mathrm{SO}}|e_{o}\uparrow\rangle\langle
e_{o}e_{v}|H_{\mathrm{VO}}|g_{0}g_{v}\rangle|}{E_{\mathrm{orb}}}.$ (65)

We start with the artificial spin-orbit coupling (also termed synthetic in the
literature [2]), that is caused by the presence of transverse magnetic field
gradients. For the gradients of the order of $(\Delta B/\Delta
x)=0.1\,\mathrm{mT/nm}$, we have
$|\bra{g_{o}\downarrow}H^{\mathrm{art}}_{\mathrm{SO}}\ket{e_{o}\uparrow}|\sim
0.2\,\mu$eV. Note that we can afford to reduce the gradient magnetic field by
one order of magnitude compared to e.g. Ref. [5], since in a quantum computing
architecture based on shuttling the QD displacement for electric dipole spin
resonance can be increased to $\sim 10$ nm, such that no compromise in the
Rabi frequency is expected. The valley-orbit coupling element can be computed
as in Sec. VII.4, and its order of magnitude can be conservatively estimated
to be comparable to the valley splitting $\langle
e_{o}e_{v}|H_{\mathrm{VO}}|g_{o}g_{v}\rangle\approx 100\,\mathrm{\mu eV}$.
Together, the gradient dominated SOC produces the spin-valley coupling
$\Delta_{\text{s-v}}^{\text{art}}\leq\,20\,\mathrm{neV}$. In absence of a
strong enough magnetic gradient, the intrinsic spin-orbit interaction gives
the same order of magnitude $\Delta_{\text{s-v}}^{\text{int}}\leq 20$ neV,
since for typical couplings ($\alpha,\beta\approx 50\,\mathrm{m/s}$ [66]),
despite larger matrix element $|\langle
g_{o}\downarrow|H^{\mathrm{int}}_{\mathrm{SO}}|e_{o}\uparrow\rangle|\sim
2\,\mu$eV, the additional reduction of $\Delta_{\text{s-v}}^{\text{art}}$ by a
factor of $E_{\mathrm{Z}}/E_{\mathrm{orb}}$ is caused by the Van-Vleck
cancellation [110, 111, 112, 113]. Next, we estimate the effective sweep rate
from the expression $a\approx E_{\text{VS}}v/L_{\text{dot}}$, which
conservatively assumes that the valley splitting $E_{\text{VS}}$ may change by
values comparable to its own magnitude over the range of a QD length scale.
Taking $E_{\text{VS}}\approx 100\,\mathrm{\mu eV}$, we can estimate $a\approx
50\,\mathrm{\mu eV}/\mathrm{ns}$ for a shuttling velocity of
$v=10\,\mathrm{m/s}$.

###  E.2 Estimation of the spin flip probability due to valley relaxation
around the hot-spot (mechanism 1)

We compute now $\delta p_{\uparrow}$ due to the mechanism of temporal spin-
valley mixing (labeled as mechanism 1). It can be done by multiplying spin
relaxation rate and the time spent around avoided crossing $\Delta_{s-v}/a$.
For a conservative estimate, we assume that the spin relaxation rate is upper-
bounded by the inverse of the valley lifetime evaluated at the Zeeman
splitting $\tau_{r,v}(E_{Z})$, and thus reads

$\delta
p_{\uparrow}^{(1)}\sim\Gamma_{s}(t)\frac{\Delta_{\text{s-v}}}{a}\geqslant\frac{1}{\tau_{r,v}(E_{Z})}\frac{\Delta_{\text{s-v}}}{a}\to\frac{0.1}{\tau_{r,v}(E_{z})[\mathrm{ns}]v[\mathrm{\tfrac{m}{s}}]}\,\,,$
(66)

where we have used the values computed above $\Delta_{s-v}^{\text{art}}\sim
200$ neV and $E_{\text{VS}}\sim 100\,\mu$eV. From Fig. 13, we conclude that
phonon-dominated relaxation yields $\tau_{r,v}(E_{z})\geq 10^{4}\,\mathrm{ns}$
for $E_{z}\leq 100\,\mathrm{\mu eV}$. Thus, $\delta p_{\uparrow}^{(1)}\leq
10^{-5}/v[m/s]$ which is at least order of magnitude smaller then a spin-flip
caused by an adiabatic transition (See Eq. 34). We point out that other
mechanisms of relaxation can dominate over phonon emission at smaller $E_{Z}$.
However, those smaller values of $E_{Z}\sim 10$ $\mu$eV would most likely
eliminate the occurrence of hotspots in the first place. Finally, let us note
that the above value most probably overestimates the relaxation experienced by
an electron going through the hotspot anticrossing in mostly diabatic way. We
have taken the maximum relaxation rate from a spin-valley hybridized lower-
energy state at the anticrossing, and while the electron spends the time
$\Delta_{s-v}/a$ in state of this character during the mostly adiabatic
evolution, its time-averaged state is probably closer to being non-hybridized
for the mostly for the mostly diabatic evolution.

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