# Blueprint of a scalable spin qubit shuttle device for coherent mid-range
qubit transfer in disordered Si/SiGe/SiO2

Veit Langrock  JARA-FIT Institute for Quantum Information, Forschungszentrum
Jülich GmbH and RWTH Aachen University, Aachen, Germany  Jan A. Krzywda
Institute of Physics, Polish Academy of Sciences, Warsaw, Poland  Niels Focke
JARA-FIT Institute for Quantum Information, Forschungszentrum Jülich GmbH and
RWTH Aachen University, Aachen, Germany  Inga Seidler  JARA-FIT Institute for
Quantum Information, Forschungszentrum Jülich GmbH and RWTH Aachen University,
Aachen, Germany  Lars R. Schreiber<EMAIL_ADDRESS>JARA-
FIT Institute for Quantum Information, Forschungszentrum Jülich GmbH and RWTH
Aachen University, Aachen, Germany  Łukasz Cywiński<EMAIL_ADDRESS>Institute of Physics, Polish Academy of Sciences, Warsaw, Poland

(today)

###### Abstract

Silicon spin qubits stand out due to their very long coherence times,
compatibility with industrial fabrication, and prospect to integrate classical
control electronics. To achieve a truly scalable architecture, a coherent mid-
range link that moves the electrons between qubit registers has been suggested
to solve the signal fan-out problem. Here, we present a blueprint of such a
$\approx 10\,\mu$m long link, called a spin qubit shuttle, which is based on
connecting an array of gates into a small number of sets. To control these
sets, only a few voltage control lines are needed and the number of these sets
and thus the number of required control signals is independent of the length
of this link. We discuss two different operation modes for the spin qubit
shuttle: A qubit conveyor, i.e. a potential minimum that smoothly moves
laterally, and a bucket brigade, in which the electron is transported through
a series of tunnel-coupled quantum dots by adiabatic passage. We find the
former approach more promising considering a realistic Si/SiGe device
including potential disorder from the charged defects at the Si/SiO2 layer, as
well as typical charge noise. Focusing on the qubit transfer fidelity in the
conveyor shuttling mode, we discuss in detail motional narrowing, the
interplay between orbital and valley excitation and relaxation in presence of
$g$-factors that depend on orbital and valley state of the electron, and
effects from spin-hotspots. We find that a transfer fidelity of 99.9 % is
feasible in Si/SiGe at a speed of $\sim$10 m/s, if the average valley
splitting and its inhomogeneity stay within realistic bounds. Operation at low
global magnetic field $\approx 20$ mT and material engineering towards high
valley splitting is favourable for reaching high fidelities of transfer.

##  I Introduction

Quantum computing architecture based on gated semiconductor quantum dots (QDs)
promises the necessary number of qubits for the use of quantum error
correction due to their very good coherence properties and direct
compatibility with established semiconductor technology [1, 2]. By using
nuclear spin-free 28Si [3], the silicon-based spin qubit dephasing time is
significantly increased, as it is no longer limited by hyperfine interaction
with remaining 29Si, but by charge noise, which couples to the spin via local
magnetic field gradients [4, 5, 6, 7]. The fidelity of single-qubit [8, 9, 10,
5] and two-qubit gates [11] already exceeds the quantum error correction
threshold [12]. Simple two-qubit gates require an overlap of the electron
wave-function [13, 14, 15], which would require a very dense two-dimensional
qubit matrix, in which topological quantum error correction can be realized
[16, 17]. However, this dense matrix approach is not easily scalable, if all
gates forming the QDs must be individually controllable. The size scale for
the outgoing signal lines and their control electronics exceeds that of the
dense qubit field by far, and leads to a signal fan-out problem [18]. One part
of the solution are multi-layer crossbar architectures [19, 20, 21], in which
individual qubits are addressed by the combination of signals at the gates’
crossing, or alternatively continuously driven qubits controlled by the global
magnetic field [22, 23]. Such dense qubits registers can be connected by
coherent links providing a two-qubit operation at a distance of approximately
10 $\mu$m, in order to make space for vias or tiling with cryoelectronics [18,
24, 25]. Coulomb interaction alone is too weak for long-range high-fidelity
two-qubit coupling [26]. First successes could be achieved by an indirect
interaction via a mm-long electromagnetic cavity [27, 28, 29, 30, 11], but
tuning of the qubit-carrying double quantum dots (DQDs) to the resonance
frequency of the cavity is challenging [30]. Off-resonant driving
theoretically circumvents this, but requires longer operation times [31]. In
addition, the fabrication of the cavities is hardly compatible with industrial
gate-fabrication.

Another method for a medium-range coupling distance of the order of 10 $\mu$m
is the controlled shuttling of the electron [32, 33], carrying the quantum
information in its spin degree of freedom, using a series of gates [34, 35].
Recently, this approach was integrated in the blueprint of a sparse spin qubit
array compatible with industrial fabrication without providing details on
control and spin coherence of the shuttling process [25]. Alternatively, the
shuttling can be used to distribute entangled pairs of electrons to distant
arrays (cores), in order to provide coherent communication between them [36].
The charge of a single electron has already been transferred in Si/SiGe over a
distance of nine tunnel-coupled QDs [37] using Landau-Zener charge transitions
[38, 39, 40, 34]. In GaAs, the spin-coherent transfer [41, 42, 43] that also
preserves spin entanglement [44] has already been shown. Some GaAs electron
conveyors employ surface acoustic waves, replacing the need for a gate array
[45, 46, 44], but velocity of shuttling with surface acoustic waves is fixed,
limiting flexibility of operation, and furthermore GaAs lacks nuclear spin-
free isotopes, making spin qubit decoherence very hard to reduce [47]. The
demonstrated Si-based shuttlers require individual tuning of the gate array to
compensate local potential disorder. Hence, the number of signal lines is
proportional to the length of the shuttler, and thus it is not solving the
fan-out problem.

We want to realize a mid-range coherent link by shuttling the electron, the
spin of which constitutes the qubit, across a distance of approximately
$L_{\mathrm{s}}=10$ $\mu$m and will refer to it as a spin qubit shuttle
(SQuS). The SQuS has to fulfil the following criteria: (I) In order to solve
the fan-out problem, the number of input terminals required has to be
independent of the length of the SQuS. A scalable quantum computer
architecture can be implemented by such a SQuS. (II) The electron transfer has
to be spin-coherent with a sufficiently low error of $\delta C\lesssim
10^{-3}$, in order to preserve the quantum information to the degree necessary
for achieving fault-tolerance using quantum error correction codes [48, 49],
or for executing NISQ algorithms [50]. (III) The transfer process has to be at
least as fast as the typical timescales of single qubit and near-range two-
qubit gates or qubit readout, in order to avoid the situation in which it is
the qubit shuttling that determines the quantum algorithm runtime. Thus, a
transfer velocity $v\sim 10$ m/s is sufficient. If shuttling is relatively
rare compared to qubit manipulation and qubit readout, an order of magnitude
lower $v$ might be feasible as well. The ratio of occurrences of these events
will depend on details of a quantum computer architecture.

We present in this paper a blueprint of such a SQuS: Scalability is achieved
by electrically connecting control lines not to individual gates, but to a few
so-called “gate sets”, where all gates within one set are electrically
connected and thus on the same potential. We discuss two distinct transport
modes: The first - the “bucket-brigade” (BB) mode - relies on periodic
modulation of voltages controlling relative detunings between adjacent QDs in
a pre-existing chain of $N\approx 100$ tunnel-coupled QDs [37]. The second -
the “conveyor belt” (CB) mode - relies on electrostatic creation of a single
deep quantum dot that is moving along a one-dimensional channel [51]. We argue
that the BB mode is less robust than the CB mode, when scalability of the
quantum computing architecture is seriously taken into account in presence of
realistic electrostatic disorder. We focus thus on the theory of shuttling in
the CB mode: we carry out theoretical optimization of the design of the
Si/SiGe structure with gate sets that predicts robust dynamics of an electron-
containing QD moving across a disordered channel, and calculate spin qubit
decoherence as a function of shuttling velocity. The main result of the paper
given is that for realistic parameters ($T_{2}^{*}$ times, orbital excitation
energies, valley splitting, density of atomic interface steps) of Si/SiGe
structures, we predict the existence of an optimal electron velocity
$v_{\mathrm{opt}}$ in the CB transfer mode. This $v_{\mathrm{opt}}$ is between
five and a few tens of m/s, and we predict that the operation of the SQuS with
this velocity will lead to qubit coherence error below the targeted $10^{-3}$,
showing that using of the proposed mid-range link will allow for scalable
quantum computing architecture.

In the following, we summarize our considerations on the spin dephasing
mechanism in the SQuS starting from the lowest qubit transfer velocities. When
the voltages controlling the SQuS are varied slowly enough, the qubit transfer
should be adiabatic, i.e. the electron should remain in its lowest-energy
orbital/valley state while it is being pushed along the channel. As for the
spin degree of freedom, if we assume that the electron does not pass through
spin-relaxation hotspots that occur when the spin splitting matches the valley
splitting in a given QD [52, 53, 54, 55, 56], the targeted transfer time,
$\tau\leq 10\,\mathrm{\mu}$s, is at least three to four orders of magnitude
below spin relaxation times in stationary dots in the presence of a magnetic
field gradient [55, 56], and the latter are not expected to be lowered
significantly due to the quantum dot moving at velocities $v\leq 100$ m/s
[33]. Note that a relatively spatially uniform valley splitting is helpful for
choosing a global magnetic field $B$ that leads to such avoidance of the hot-
spots.

In absence of the hot-spots, spin state can then only undergo dephasing due to
fluctuations of local values of spin splitting along the channel due to
nuclear dynamics [57] and charge noise [58, 59, 60, 61, 62, 6, 7, 63]
modulating the effects of spin-orbit interactions on the spin splitting of an
electron (e.g. through fluctuations of $g$-factors) in a QD at a given
location. For a stationary QD, these fluctuations lead to finite $T_{2}^{*}$
dephasing times [64, 5, 65, 66, 6], and the motion of the electron through a
channel longer than the correlation length of random contributions to spin
splitting enhances the spin dephasing time due to the motional narrowing
effect [57]. Making the shuttling velocity $v$ higher seems then to be an
obvious way to suppress the phase error: the qubit spends less time exposed to
perturbations, and their noisy influence is additionally suppressed by fast
motion. However, with increasing $v$, changes in electrostatic potentials and
valley fields experienced by the electron become faster, and the assumption of
adiabatic character of the evolution of its orbital and valley degrees of
freedom has to become untenable.

When the dynamics of the electron becomes non-adiabatic, motion-induced
excitation of the electron into higher-energy states has to be taken into
account. Transitions to excited orbital states of the electron in a potential
of the moving QD are caused by electrostatic disorder in the channel: in the
frame co-moving with the QD the quasi-statically fluctuating disorder turns
into dynamic noise coupling the orbital levels. Analogously, atomic-scale
interface roughness [67, 68, 1] that affects the valley splitting [69, 70, 64,
71, 72, 28, 13, 73, 74, 55, 56, 52, 54, 75, 76] and determines the composition
of valley states [67, 77, 68, 56, 78, 79] in a static QD, becomes a time-
dependent valley-coupling term for a moving QD, with intervalley excitations
appearing as a result. Once the electron starts to occupy excited states, it
becomes susceptible to processes of energy relaxation accompanied by emission
of phonons. This makes the evolution of the orbital/valley state stochastic:
the electron will spend random fractions of shuttling time in various
orbital/valley states, thus opening up a new channel for qubit dephasing.
Spin-orbit coupling makes the $g$-factor state dependent [64, 80, 81], with a
relative variation of electron spin $g$-factor between distinct valley states
being $\sim 10^{-3}$ [64, 73]. A similar $g$-factor difference was measured
between neighouring QDs [82, 83], which we assume to be an upper bound for
g-factor difference between lowest-energy and excited orbital in a single QD.
$B$-field independent contribution to spin splitting due to spin-orbit
interaction also depends on the valley state [84, 66]. Consequently, any
randomness in time spent by the electron in distinct valley/orbital states
will lead to randomness in qubit phase (and thus dephasing) for any finite
external magnetic field. With rms of the phase given by $\delta\phi$, and the
probability of excitation out of instantaneous ground state given by $p_{e}$,
the phase error $\delta C$ is given by

$\delta C=p_{e}(1-e^{-\delta\phi^{2}/2})\approx
p_{e}\delta\phi^{2}/2\,\,\mathrm{when}\,\delta\phi\ll 1\,\,,$ (1)

where in the first formula we have assumed that the distribution of the random
contributions to qubit phase is Gaussian, while the second one, relevant in
the small-error regime of interest here, does not require this assumption. As
$\delta C\\!\leq\\!p_{e}$, limiting the probability of orbital and valley
excitations, i.e. keeping $p_{e}\\!<\\!10^{-3}$, is one route towards reaching
the targeted level of phase error. When this turns out to be impossible,
suppression of $\delta\phi$, e.g. by making the orbital/valley relaxation
faster (thus making the electron spend shorter periods of time in excited
states), is the remaining route towards a coherent SQuS. Quantitative
calculation of both orbital/valley excitations caused by a QD motion, and
orbital/valley relaxation, as functions of parameters of the SQuS, is thus the
main topic of the second part of the paper, in which we focus on coherence of
the electron transferred in the CB mode.

Figure 1: A sketch of the transfer errors of an electron spin-qubit after 10
$\mu$m shuttling in the CB mode as a function of the shuttling velocity for
various decoherence mechanisms discussed in this paper. In Sec. VIII, we
discuss the detailed plots of these dependencies that summarize all the
calculations in this paper.

The key results of our calculations are the following: Already for reliable
transfer of the electron, the CB mode of electron-spin shuttling is superior
vs. its BB counterpart in terms of scalability, i.e. robustness against
potential disorder with only few signal lines independent from the SQuS-length
available (Sec. II). The propagating QD potential required for the CB mode can
be generated in a realistic Si/SiGe SQuS despite having typical density of
charged defects at the Si/SiO2 interface (Sec. III). Then, motional narrowing
enhances spin coherence of a single electron confined in the propagating QD
compared to a static QD, setting a comfortable lower velocity limit to the CB-
mode (red line in Fig. 1). The state-dependent electron g-factor sets the
upper velocity limit in conjunction with diabatic QD motion (Sec. V): Orbital
excitations (black line in Fig. 1) appear rarely and relax quickly in the
propagating QD in our realistic Si/SiGe SQuS, so they do not limit the spin
coherence (Sec. VI). However, valley excitations last orders of magnitude
longer, setting the upper velocity limit of the CB (green and blue lines in
Fig. 1) within the bounds of our models of the lateral valley-splitting
fluctuations (Sec. VII). Spin relaxation hot-spots have to be passed
sufficiently fast, setting another lower velocity limit (yellow line in Fig.
1) (Sec. VII). Finally, dotted line shows that motion-induced spin relaxation
caused by spin-orbit interaction analyzed previously in [33] (see Sec. VI.3)
does not pose a threat to the coherence of the shuttled qubit. Taking all
spin-decoherence mechanisms into account in Sec. VIII, we predict that CB mode
shuttling across a distance of 10 $\mu$m with less than 0.1 % infidelity is
feasible under favorable QD velocity (white area underneath gray line in Fig.
1), magnetic field, and SQuS-geometry.

##  II SQuS device concept

Figure 2: SQuS concept and operation modes. (a) Sketch of a SQuS device.
(b)-(g) Comparison of BB (left panels) and CB (right panels) transfer modes.
(b),(e): Position of the clavier gates and electric connection scheme for the
BB mode (b) and CB mode (e), respectively. (c),(f): Input voltages during a
SQuS step cycle T for the BB mode (c) and CB mode (f), respectively. d,g:
Electrical potential (blue) calculated along the center of a SQuS within the
Si/SiGe quantum well for four time frames (dashed lines in panels b and d for
each mode). Dashed red line indicates the calculated electron density.

A sketch of the SQuS device is shown in Fig. 2a. It is based on multi-layer
electrostatic gates and thus its fabrication is compatible with common QD
devices [37, 52], in which qubit manipulation with necessary fidelities was
demonstrated, and can be readily adapted by industrial fabrication lines [85].
The shuttling process is controlled by an array of metal gates called clavier
gates on top of a Si/SiGe heterostructure. Two parallel gates (called
screening gates) underneath the clavier gate layer, define a one-dimensional
electron channel and screen the electric field of the clavier gates at the
edge of the channel. The DC voltage applied to these gates is chosen to
deplete this channel. Our main idea is to connect a few control lines to a
small number of gate sets (i.e. clavier gates set to the same potential), the
number of which is independent of the length of the coherent link [51]. The
constant number of input terminals of the coherent link solves the signal fan-
out problem and assures full scalability of our approach. Along the SQuS
channel we require no charge detector. For a SQuS test device, we suggest two
single electron transistors (SET) at each end of the channel, which detects
the charge state at the end of the channel (Fig. 2a). If the SET is tunnel-
coupled to the channel, electrons can be loaded into and unloaded from the
channel on demand, controlled by small number of dedicated gates at the end of
the SQuS. For a quantum computing architecture, other approaches for
loading/detecting electrons may be found.

###  II.1 Two modes of qubit transfer

There are two modes of operation for a shuttling which we call “bucket
brigade” (BB) and “conveyor belt” (CB) transfer mode, which differ by the
connection scheme of the clavier gates (Fig. 2b for BB and 2e for CB,
respectively). Both modes are explained in more detail below.

The BB transfer mode (Fig. 2, left panel column) requires a linear array of
QDs along the SQuS, which can be pairwise described by the Hamiltonian
$H_{k}=\epsilon_{k}\sigma_{z}/2+t_{c,k}\sigma_{x}$, where $\epsilon_{k}$ and
$t_{c,k}$ are the interdot energy detuning and interdot tunnel-coupling of the
$k$-th double quantum dot (DQD) pair of the linear array, respectively, and
$\sigma_{x}$ and $\sigma_{z}$ are Pauli-matrices. The electron is transferred
by adiabatic Landau-Zener transitions (LZT) between adjacent DQDs: time-
dependent voltages applied to adjacent QDs change $\epsilon_{k}$ from negative
to positive, and the energies of states localized in each of the two QDs anti-
cross, with the minimum gap given by twice the tunnel coupling $t_{c,k}$ (Fig.
3a). Three clavier gate sets (1,2,3) operating as plunger gates (i.e.
controlling mainly the chemical potential of the QD underneath) in Fig. 2b
trigger the LZTs at time $D/2$ by sweeping the gate voltages of the gates sets
as plotted in Fig. 2c. The potential at four time frames is sketched in Fig.
2d. The channel is initially depleted and only one QD is filled (red
wavefunction in Fig. 2d) by loading a single electron from one channel end.

In the CB mode (Fig. 2, right panel column), the existence of an array of
tunnel-coupled QDs is not needed. Every fourth clavier gate is connected (Fig.
2e) and by applying sine-signal with $\pi/2$ phase shift to each gate set as
depicted in Fig. 2f, a propagating sine-wave potential in the 1D quantum
channel is induced (Fig. 2g) with only one pocket filled by the electron to be
transferred (red wavefunction in Fig. 2g). We refer to this pocket as single
moving QD. Tunnel coupling between pockets of the sine function has to be
excluded by proper choice of gate pitch and the signal amplitude. We will show
that this requirement can easily be fulfilled by a realistic device in Section
III, for which electron shuttling on a short length scale has been
demonstrated already [51]. A minimum of three gate sets is required to define
the direction of the transfer. We choose a 4-gate sets scheme here, since it
eases the realisation of a connection scheme of the SQuS in the CB mode as
will be elaborated in Section III. The input signals sketched in Fig. 2c,f
presume a constant transfer velocity. Smooth acceleration can be implemented
in CB by sweeping the frequency of the SQuS signals applied to the clavier
gates. Adiabatic reversal of the transfer direction is feasible as well. Such
a flexibility is lacking in the surface acoustic wave approach demonstrated in
GaAs as the speed of sound is intrinsically fixed. Note that mixtures of the
BB and CB modes with more input signals might be beneficial, but we restrict
the considerations to these two extreme case of transfer modes.

Figure 3: Schematic picture of the energy spectrum during transfer of spin
superposition in presence of orbital (ground and first excited only), valley
and spin degrees of freedom, as a function of: (a) DQDs detuning $\epsilon$ in
a single transition between the QDs in Bucket Brigade, (b) position along the
SQuS in the Conveyor Belt. We denoted excited valley states $\ket{e_{v}}$
using dashed line and the joint ground state of orbital and valley degrees of
freedom $\ket{g_{o,v}}$ by thicker: blue (spin down) and red (spin up) line.
In the insets, we illustrate mechanism of electron shuttling in each mode,
note that contrary to BB mode the charge transfer in CB does not require
effectively adiabatic evolution in the lowest energy levels. In the conveyor
belt (b), we plotted energy difference between excited and ground orbital
levels including expected electrostatic disorder (See III.2). We also included
the modification of energy spectrum due to atomistic steps (left kink) and a
smooth interface gradient (right kink), that lead to temporal reduction of
valley splitting $E_{\text{VS}}$.

###  II.2 Challenges in the bucket-brigade mode

Within our scalable gate-set approach two significant limitations arise in BB
transfer mode, both originating from typical potential disorder in Si devices:
Firstly, fine-tuning of gate voltages for setting the $t_{c,k}$ close to a
common value of $t_{c}$, and secondly fine-tuning of voltage pulses applied to
specific QDs triggering adiabatic LZTs along the chain, are both impossible to
achieve, since gates are electrically connected. As potential disorder is
unavoidable, we have to deal with a range of $t_{c,k}$, while the sweeping
range for $\epsilon_{k}$ at all LZT has to be large enough to compensate
offsets in zero-detuning points among adjacent QDs. The requirement of being
able to transfer the qubit with the use of only a few synchronized time-
dependent voltage signals (Fig. 2c), applied to appropriately designed gates
spanning the whole length of the SQuS, leads to rather stringent requirements
on the degree of uniformity of the channel through which the electron is to be
sent. In the following we quantitatively explicate on this requirement.
Considering a 10 $\mu$m long SQuS, an array of $N\\!\sim\\!100$ tunnel-coupled
QDs is required to span the distance. Then the qubit shuttling is effected by
consecutive LZT of the electron between neighboring DQDs, driven by a sweep of
the DQD interdot detuning $\epsilon$ (Fig. 3a). First, we focus on the
probability $Q$ of nonadiabatic LZT evolution given by $Q_{k}\\!=\\!\exp(-2\pi
t_{c,k}^{2}/\hbar v_{\epsilon})$, where $v_{\epsilon}$ is the rate of change
of detuning [86]. This $Q_{k}$ is the probability that the electron will fail
to transfer adiabatically from QD $k$ to QD $k+1$ when $\epsilon_{k}$ is
driven through the anticrossing of tunnel-coupled states localized in the two
QDs. It should be stressed that while in principle the charge transfer could
occur in an inelastic way (via phonon-assisted tunneling) in subsequent part
of the BB driving cycle in which the energy of electron in $k$-th QD is larger
than the energy in $k+1$-th QD (between $t\\!=\\!D/2$ and $t=D$, see Fig. 2),
this process is inefficient in Si/SiGe QDs [87], and it can be neglected on
timescales relevant here.

To achieve adiabatic (and thus deterministic) evolution through the whole
length of the SQuS with an error of $<10^{-3}$, each of the $N$ transfers has
to fulfill $Q<10^{-5}$. This condition is rather restrictive as can be
understood by translating it into a lower limit of all $t_{c,k}$: Let us
assume that the transfer time $L_{\mathrm{s}}/v$ through the SQuS is at least
10 $\mu$s, to avoid limitation of quantum computer clock speed. For $N\approx
100$ this means that the interdot transfer time has to be $\approx\\!100$ ns.
In state-of-art Si/SiGe devices, the offsets in zero-detuning points among
adjacent QDs have a Gaussian distribution with an rms of about 3 meV (cf.
simulations in Fig. 9). Thus the potential disorder sets rigorous restrictions
on $\epsilon_{k}(t)$: (I) It has to span a range of $\approx 10$ meV in order
to include all $N-1$ DQD zero-detuning points (at time $D/2$ in Fig. 2d),
since no individual compensation of disorder is possible in our gate-set
approach. (II) $v_{\epsilon}$ has to be constant, since detuning at which
tunneling-induced anticrossing of states occurs is unknown due to disorder.
(III) $v_{\epsilon}>100$ $\mu$eV/ns to pass the SQuS in less than 10 $\mu$s.
This conditions translate into $t_{c,k}>10$ $\mu$eV for all $N-1$ DQD-pairs,
in order to achieve adiabatic charge transfer across the whole SQuS.
Accordingly, passing the SQuS in less than 1 $\mu$s, requires $t_{c,k}>35$
$\mu$eV. We believe that such a high $t_{c,k}$ is hard to achieve for all $k$
with state-of-art disorder in Si/SiGe devices, if only a common voltage can be
applied to all the barrier gates. In particular, for an ensemble of $N\sim
100$ QDs with expected value of $t_{c}=35\,\mu$eV, the $35\,\%$ variation of
tunnel coupling would on average result in two weakly coupled QD pairs (with
$t_{c}<10\,\mu$eV) for which adiabatic transfer would fail with high
probability.

Figure 4: Illustration of the propagation of an electron in BB mode for which
the QD transfer fails once (red) and comparison with an electron which follows
the intended path (blue). (a) Three input signal as a function of time applied
to the gate sets labeled 1, 2 and 3 in panel b. (b) Labeling of the clavier
gates and electrical connection. Fixed voltages are applied to all barrier
gates, labeled 0. (c) Sketch of the potential in the QW along the SQuS channel
aligned with the clavier gates in panel b. The potential plotted in each sub-
panel corresponds to a different time $t$ labeled in panel a. Blue circles
show the intended position of the shuttled electron and red circles mark the
expected electron position in case of a failed LZR transition marked by the
circle in panel a.

How does imperfectly adiabatic LZT degrade qubit coherence? Naively, one might
think the qubit will arrive just a few BB signal cycles delayed. Besides
besides losing track of the exact qubit position this will lead to qubit
dephasing, if the Zeeman energies of all QDs are not exactly equal due to
small variations of electron g-factor and/or local magnetic field. In fact, it
can become much worse, since only one diabatic LZT might lead to a reversal of
the shuttling direction as illustrated in Fig. 4. If the adiabatic transfer is
unsuccessful within the time range 0 to $D_{1}$ (electron position marked
red), then the electron will move backward, if the following LZT within time
$D_{1}$ to $T_{1}$ is adiabatic. While this following LZT can be to some
extent enforced to be diabatic by a fast detuning ramp, this is not possible
for the LZT transition at $C_{3}$. The effect is that the electron starts to
shuttle in the opposite direction as can be seen by comparing the electron
position marked blue (intended) and red (unintened) in Fig. 4. As noted
before, electron-phonon coupling in Si is too weak to allow for efficient
enough inelastic phonon-assisted tunneling between detuned QDs [87] and thus
this back-transfer can be persistent for several BB signal cycles. Such an
unintentional reversal of shuttling direction is catastrophic to a quantum
computing architecture and can be triggered by only one diabatic LZT of two
adjacent QDs. This rules out the BB transfer mode, unless ways to mitigate
this reversal are found. Certainly, lowering the potential disorder will lead
to significant improvements of stability of operation, as it will allow for
smaller sweep range (and thus lower $v_{\epsilon}$ while keeping the same
total shuttling time). There are even more challenges of the BB transfer mode:
If the LZT evolution is made very slow to ensure low $Q$, not only transfer
velocity $v$ and thus clock-speed sets a limit, but processes of electron
excitation due to charge noise become significant, and in fact at lower $v$
their presence limits $Q$ from below, as it has been shown in recent work
considering coupling of the transferred electron to phonons and sources of
both $1/f$ and Johnson-Nyquist charge noise [40, 87]. Additionally, transition
between neighbouring sites separated by $50-100\,\mathrm{nm}$ might lead to
valley excitations, caused by spatial variation of valley splitting
$E_{\text{VS}}$ in typical Si/SiGe heterostructures (Fig. 3a) [88]. The
temporal occupation of higher valley and presence of valley-orbit mixing can
lead to spin decoherence as discussed in detail in this paper in the context
of the CB mode.

We conclude that coherence error below $10^{-3}$ will be difficult to achieve
in scalable BB mode without a significant improvement of the uniformity of
state of the art devices. The tension between conflicting requirements of
small interdot barriers resulting in large tunnel couplings (necessary for
deterministic, and consequently coherent spin shuttling) and homogeneity of
parameters of 100 QDs in a realistically disordered heterostructure (necessary
for scalability) is absent in the CB mode of operation. In the remaining part
of the paper we will thus focus on analysis of that mode of the SQuS.

###  II.3 Larger robustness of the conveyor-belt mode

In order to start thinking about qubit transfer in CB mode, we only have to
require that the depth of the single moving QD is much larger than than a
typical variation of electrostatic disorder potential on length-scale of QD
size. Compared to the BB mode, we do not have to worry about tension between
the requirement for existence of separate (albeit well-coupled with finite
$t_{c}$) quantum dots, and the need for charge-transfer control with global
pulses, both in presence of disorder. We only need to create a moving QD of a
stable shape (Fig. 3b).

Due to this, we expect the CB mode to be more robust to disorder in the
channel as will be further discussed in Section III. Considering signal
generation and bandwidth of signal lines, a maximal input signal frequency of
100 MHz is convenient, and it results in a sufficiently high transfer velocity
of 20 m/s at a typical gate pitch of 50 nm. As a disadvantage compared to BB
mode, the CB mode requires a higher dynamic range of the input signals, which
might cause Ohmic heating, if the clavier gates are not superconducting. It
also sets limits on the distance between clavier gates and the moving QD, and
thus the depth of the QW and the clavier gate pitch must be well-balanced
(Section III). Transfer velocity, signal amplitude and other issues such as
spatial fluctuations of $E_{\text{VS}}$ (shown in Fig. 3b) affecting the
coherent spin transfer in the CB mode will be discussed in the following
sections.

##  III Optimization of gate-design for conveyor belt mode shuttling

In this Section we elaborate further on the blueprint of a SQuS operating in
conveyer belt (CB) mode and optimize the gate design of a realistic undoped
Si/SiGe SQuS for CB mode, as the upper SiGe spacer layer keeps charged
defects, typical for the semiconductor-oxide interface, at a larger distance
and thus reduce its impact on potential disorder [89]. In order maximize the
robustness of the adiabatic charge shuttling against the potential disorder,
we check whether its magnitude in the SQuS channel is sufficiently low
compared to the confinement of the propagating QD and whether the associated
correlation length of disorder is sufficiently large not to break the QD
apart. As the dominant source for potential fluctuations, we simulate the
impact of charged defects at the interface between a thin Si cap layer (on top
of the Si/SiGe heterostructure) and the planar SiO2 layer. The model employed
for finite-element calculations is based on realistic Si/SiGe device with
three metallic gate layers (Fig. 5a). By alignment of two clavier gate layers
fabricated by electron-beam lithography, we can achieve an effective minimal
gate pitch of $l_{\mathrm{g}}+l_{\mathrm{ox}}=$35 nm, where $l_{\mathrm{g}}$
and $l_{\mathrm{ox}}$ are the width of a single clavier gate and the oxide
thickness, respectively. The connection scheme of the clavier gates required
for CB mode (cf. Fig. 2) can be realised within each of the two clavier gate
layers by connections on both sides of the SQuS (Fig. 5a). We will discuss
whether this minimal gate pitch is sufficient for the SQuS or whether even
larger gate pitches are optimal and thus fabrication constraints can be
relaxed. For finite element simulations of the SQuS electrostatics, we use a
COMSOL model of the realistic SQuS device, a part of which is shown in Fig.
5b. In this model having a total size of 2600 nm by 400 nm, we randomly
distribute spatially uncorrelated singly charged defects at the planar Si/SiO2
interface having a density of $5\times 10^{10}$ cm-2. This is a typical defect
density for Si/SiO2 interface extracted from room-temperature C-V measurements
[90, 91, 92].

Figure 5: Realistic Si/SiGe SQuS device. (a) Scanning electron micrograph of
10 $\mu$m long SQuS showing three gate layers: on top of a screening gate
layer (colored in green), two clavier gate layers connected to four gate sets
for CB mode. Gates at the ends of the SQuS are for the formation of single
electron transistors. (b) COMSOL model of the SQuS device from panel a showing
two SQuS unit cells and thus eight clavier gates. The typical thickness of
layers is given in brackets.

###  III.1 Analytical optimization of the device design without defects

The geometry of the SQuS is not only given by the width of the clavier gates
$l_{\mathrm{g}}$ and their spacing $l_{\mathrm{ox}}$, but also by the
thicknesses $h_{\mathrm{ox}}$, $h_{\mathrm{SiGe}}$ and $h_{\mathrm{Si}}\approx
10$ nm of the oxide layer, the Si0.7Ge0.3 layer and the thickness of the
strained Si quantum well (QW) layer, respectively, as well as the spacing
between the screening gates $w_{\mathrm{sp}}$ (Fig. 5b). As the parameter
space for the SQuS geometry is large, we start our discussion by neglecting
the charged defects and investigate the harmonicity of the propagating
potential in the center of the quantum well layer as a function of the
interplay between the depth of the quantum well and the gate pitch. We aim at
maximizing the orbital energy along the SQuS transfer direction
$E_{\mathrm{orb}}^{x}(x)$ at a given sine wave voltage amplitude $V_{a}$
applied to the clavier gate sets at all positions of the propagating QD. For
further simplicity, we assume that all four clavier gate sets are at the same
height (other than plotted in Fig. 5b), thus the thickness of the SiO2 layer
$h_{\mathrm{ox}}$ underneath each layer is the same. This approximation allows
us to argue with the Fourier analysis of the potential formed by the array of
clavier gates, in order to find an analytical expression for the relation of
the gate pitch to the depth of the QW given by the thickness of the oxide and
the thickness $h_{\mathrm{SiGe}}$ of the SiGe top layer (a 1 nm thin Si cap
layer is neglected). We assume $w_{\mathrm{sp}}$ to be large enough that we
can reduce the problem to the xz-plane. As the SQuS potential exhibits a
periodicity of four gate pitches due to the use of four gate sets, we
introduce the unit cell length of the SQuS
$L_{x}=4(l_{\mathrm{g}}+l_{\mathrm{ox}})$. For a homogeneous dielectric, we
then find an optimal unit cell length of
$L_{x}=L_{\mathrm{opt}}\equiv\pi(h_{\mathrm{SiGe}}+h_{\mathrm{ox}})$, with
which the orbital confinement energy along the $x$-direction scales as
$E_{\mathrm{orb}}\propto\frac{1}{L_{\mathrm{opt}}}$, inversely proportional to
the depth of the quantum well. Fig. 6a compares the estimated optimum
$L_{\mathrm{opt}}$ to a numerically exact solution of the one dimensional
potential. Details on the calculations and additional information are given in
Appendix A.

Figure 6: Orbital-splitting as a function of the geometric parameters of the
SQuS without disorder. (a) Using $h_{\mathrm{ox}}=10$ nm and $V_{a}=100$ mV,
the minimum orbital splitting $E_{\mathrm{orb}}^{x}$ is computed by solving
the Schrödinger equation on the full-mode periodic potential. The red dashed
line indicates $L_{\mathrm{opt}}$ and the red solid line the optimal unit cell
length for the case of inhomogeneous dielectics
$L_{\mathrm{opt}}^{\mathrm{inh}}$ (discussed in Appendix A). (b) Orbital
splitting $E_{\mathrm{orb}}^{x}$ (filled symbols) and $E_{\mathrm{orb}}^{y}$
(open symbols) as a function of the gap between the screening gates
$w_{\mathrm{sp}}$ for an oxide thickness $h_{ox}=5$ nm (blue symbols) and
$h_{\mathrm{ox}}=10$ nm (orange symbols) and the voltage $V_{S}$ applied to
the screening gates (see the model in Fig. 5b). A fixed sin-signal amplitude
$V_{a}=100$ mV is applied to the clavier gates and $h_{\mathrm{SiGe}}=45$ nm.

Next, we analyse the effect of $w_{\mathrm{sp}}$ on the curvature of the QD
potential by harmonic fits in $x$\- and $y$-direction and express the result
as an effective $E_{\mathrm{orb}}^{x}$ and $E_{\mathrm{orb}}^{y}$,
respectively. We plot the minimum of all $E_{\mathrm{orb}}^{x}(x)$ and
$E_{\mathrm{orb}}^{y}(x)$ of all $x$-positions along the SQuS unit cell for
two oxide thicknesses (Fig. 6b). As $w_{\mathrm{sp}}$ is increased, the
capacitive coupling of the clavier gates to the QD increase and therefore
$E_{\mathrm{orb}}^{x}$ increases with $w_{\mathrm{sp}}$. For larger
$w_{\mathrm{sp}}$, the gain in $E_{\mathrm{orb}}^{x}(x)$ decreases. On the
other hand, $E_{\mathrm{orb}}^{y}(x)$ decreases with increasing
$w_{\mathrm{sp}}$, since with widening of the SQuS channel the QD becomes more
elliptical towards the $y$-direction. The QD confinement is maximized, if the
QD remains approximately circular during the shuttling, which is fulfilled at
$w_{\mathrm{sp}}\\!\approx\\!200$ nm here. A negative voltage applied to the
screening gates $V_{S}$ can be used as an additional degree of freedom to
enhance $E_{\mathrm{orb}}^{y}$ independent from $E_{\mathrm{orb}}^{x}$ (Fig.
6b).

###  III.2 Numerical simulation of the QD shuttling in presence of interface
defects

Figure 7: Simulation based on Poisson-Schrödinger solutions of propagating
potential in CB mode for 800 ensembles of randomly distributed charged defects
at the Si/SiO2 interface for $h_{\mathrm{SiGe}}=45$ nm and
$w_{\mathrm{sp}}=200$ nm. (a) Orbital energy $E_{\mathrm{orb}}$ of the first
excited state of the propagating QD assuming an oxide thickness
$h_{\mathrm{ox}}=10$ nm, gate width $l_{\mathrm{g}}=30$ nm and signal
amplitude of $V_{a,1}=45$ mV and $V_{a,2}=100$ mV applied to the first and
second claviature gate layer, respectively. (b) Same as in panel a with
parameters ($h_{\mathrm{ox}}=10$ nm, $V_{a,1}=90$ mV, $V_{a,2}=200$ mV,
$l_{\mathrm{g}}=30$ nm). Inset: The probability density
$\int|\psi(x,y)|^{2}dy$ of the ground electron wavefunction is plotted for all
expected QD positions $x_{dot}$ and all charge defect ensembles. Wavefunctions
deviating from a single QD are marked in red together with their corresponding
$E_{orb}$ data point. (c) Same as in panel b with parameters
($h_{\mathrm{ox}}=5$ nm,$V_{a,1}=60$ mV, $V_{a,2}=100$ mV, $l_{\mathrm{g}}=30$
nm) and the probability density in the inset. (d) Same as in panel b and c
with parameters ($h_{\mathrm{ox}}=5$ nm, $V_{a,1}=65$ mV, $V_{a,2}=100$ mV,
$l_{\mathrm{g}}=60$ nm) and the probability density in the inset. In addition
to the first excited orbital energy (blue dots), the second excited orbital
energy is plotted (bright blue dots). (e)-(f) The numerically calculate
velocity for each ensemble for the ground (black) and first excited state
(orange) along the SQuS direction is plotted corresponding to the parameters
from panel a and d, respectively.

We analyse now the impact of charged defects at the Si/SiO2 interface on the
SQuS in CB mode. We numerically calculate the potential within the center of
the 10 nm thick QW plane applying the model shown in Fig. 5b. The orbital
energy of the first excited state of this QD is obtained using a Poisson-
Schrödinger solver. Since the SQuS is periodic in $x$-direction, we focus on a
unit cell of the SQuS consisting of four clavier gates and having a length of
$L_{x}=4(l_{\mathrm{g}}+l_{\mathrm{ox}})$. We calculate the potential in each
unit cell 800 times with 800 different ensembles of randomly distributed
singly charged defects all having on average constant defect density of
$5\times 10^{10}$ cm-2. We focus a QW depth of $h_{\mathrm{SiGe}}=45$ nm here
and will optimize the gate pitch $l_{g}+l_{\mathrm{ox}}$ and the screening
gate spacing $w_{\mathrm{sp}}$ according to Fig. 6. We assume the oxide to be
conformally deposited and thus $h_{\mathrm{ox}}\approx l_{\mathrm{ox}}$. We
start close to our minimal clavier gate pitch of 40 nm and an oxide thickness
$h_{\mathrm{ox}}$ of 10 nm, hence $l_{\mathrm{g}}=30$ nm. The spacing between
the screening gates $w_{\mathrm{sp}}$ is fixed to 200 nm and
$V_{S}\approx-100$ mV to obtain an approximately circular propagating QD. For
calculation of the electron ground state and the first excited orbital state,
we solve the time-independent Schrödinger equation for the 2D x,y-potential in
the center of the QW at various positions $x_{n}$ along the center of the
QuBus. We solve numerically within boundaries of $\pm 0.5L_{x}$ around
$x_{n}$, in order to exclude the neighbouring minima of the periodic
potential.

Applying sine wave signals with an amplitude of $V_{a,1}=45$ mV to clavier
gates on the lower and $V_{a,2}=100$ mV to the upper gate layer, the first
orbital splitting energy $E_{orb}$ fluctuates considerably along the SQuS unit
cell and among the defect ensembles (Fig. 7a). For some defect ensembles, the
orbital splitting approaches zero. If we double the signal voltage amplitude
($V_{a,1}=90$ mV and $V_{a,2}=200$ mV) applied to the clavier gates (Fig. 7b),
the orbital splitting $E_{\mathrm{orb}}$ of the propagating QD is enlarged and
the variance is reduced. Further investigations of the wavefunction (inset of
Fig. 7b) reveal that the propagating QD breaks into a tunnel-coupled double
quantum dot (DQD) at the locations (red lines in insert of Fig. 7b). The
second QD appears either because the propagating QD splits into a DQD at a
large potential ripple or it approaches a disorder-induced deep QD. The
unintentional formation of a second QDs, which is strongly tunnel-coupled to
the propagating QD might lead to orbital excitation and therefore must be
avoided. The low $E_{\mathrm{orb}}$ values in close proximity belong to the
same defect ensemble and appear in our simulation when the randomly
distributed defects form a charged cluster. Since we have not taken
correlation effects for the distribution of charged defects into account, such
clusters are expected to be suppressed in realistic devices due their Coulomb
repulsion, but cannot be fully excluded here.

Note a large signal amplitude $V_{a,2}$ might lead to other problems such as
sample heating. Hence, just increasing the voltage amplitudes is insufficient
and geometrical optimization of the SQuS is desirable as well. Therefore, we
reduce the maximum amplitude back to $V_{a,2}=100$ mV and $V_{a,1}=60$ mV
similar to (Fig. 7a) and reduce the SiO2 thickness to $l_{\mathrm{ox}}=$5 nm.
The resulting variance of the orbital splitting energies is reduced (Fig. 7c)
compared to Fig. 7a mainly due to an enhanced screening of the defects by the
metal gates. The metal gates are also closer to the QW, which increase the
confinement of the intentional QD slightly. Still we observe the propagating
QD breaking into a DQD (red lines in insert of Fig. 7c) for some defect
ensembles.

Next, we enlarge the width of the finger gate to $l_{\mathrm{g}}=60$ nm
keeping the oxide thickness at $l_{\mathrm{ox}}=5$ nm. Such a SQuS is not only
easier to fabricate, but also exhibits a larger orbital splitting staying
above 0.93 meV for all 800 defect ensembles (Fig. 7d), since the capacitive
coupling of the gates is enhanced. The difference of the orbital energies of
the first and second exited orbital reveals only a small ellipticity of the
propagating QD. Most noteably, the electron wavefunction (insert of Fig. 7d)
shows no trace of breaking into a DQD. Thus, the QD can propagate sufficiently
smoothly across the disordered potential (cf. the results on disorder
autocorrelation function in Sec. VI). The increased gate pitch also enhances
anharmonicity of the propagating potential as expected from Fig. 6a. It
results in wobbling and breathing of the QD visible by the deterministic
increasing and lowering of the orbital splitting with a wavelength given by
the gate pitch, the magnitude of which however does not exceed the effect of
$E_{\mathrm{orb}}$ fluctuations due to the potential disorder.

We have also considered the expectation value of the $x$-position of the
propagating QD as a function time for the ground and first excited orbital. In
order to follow correlations in the variation of the QD location within one
defect ensemble, we calculate the corresponding finite velocity of the QD
orbitals (Fig. 7e,f). The velocity corresponding to the parameters used for
Fig. 7a reveal large variations in the positions of the QD (Fig. 7e). The
optimized SQuS geometry discussed in (Fig. 7d), reveals smaller variations in
velocity (Fig. 7f). The deterministic variation of the QD velocity following
the gate periodicity, is of the same order of magnitude as variations of the
QD orbital splitting and velocity variations due to potential disorder. These
simulations show that QD propagation is feasible despite realistic charge
disorder at the Si/SiO2 interface, but the shuttling velocity $v$ will not be
strictly constant. We always have to consider a variance of shuttling
velocities when we calculate the coherent spin transport conditions, see Tab.
1.

###  III.3 Conclusion of device design optimization

We have arrived at a realistic SQuS design for the CB mode. The numerical
simulations indicate that state-of-art defect densities at the Si/SiO2 are
sufficient to realize a QD transfer. Obviously, disorder can be counteracted
by increasing the dynamic voltage $V_{a}$ on the clavier gates, but this
approach is limited by Ohmic heating, which is is expected due to dielectric
loss in the oxides and Ohmic dissipation if normal-conductive metallic gates
are used. The latter will appear not directly at the SQuS as clavier gates
constitutes an open terminal. Decisive is the optimization of the SQuS
geometry: Using thin oxides with low defect density, the charged defects can
be screened and the metal coveraged increased. For each $h_{\mathrm{SiGe}}$, a
clavier gate-pitch can be chosen to maximize $E_{\mathrm{orb}}$. Increasing
the gate-pitch further leads on the one hand to breathing of the QD potential
and thus to deterministic oscillations of the orbital splitting and to
deterministic variations of the transfer velocity $v$. On the other hand, it
increases the mean orbital splitting due to an enhanced capacitive coupling.
The deterministic breathing effect can be balanced with the stochastic
$E_{\mathrm{orb}}$-variations due to disorder. The QD should be approximately
circular, which can be achieved by a proper gap between the screening gates
$w_{\mathrm{sp}}$, and the applied voltage $V_{S}$. If the QD is elliptical in
$y$-direction, $E_{\mathrm{orb}}$ is limited by a weak confinement in this
direction. If the QD is elliptical in $x$-direction, an increase of
$w_{\mathrm{sp}}$ can increase the capacitive coupling of the clavier gates
and thus deepen QD potential, hence increasing $E_{\mathrm{orb}}$.

Table 1: Relevant parameters and their ranges of values, which will be used for calculation of spin dephasing during CB mode type shuttling in the following sections. Parameter | low | usual | high
---|---|---|---
Orbital energy $E_{\mathrm{orb}}$ [meV] | 1 |  | 3
QD size $L$ [nm] | 12 |  | 20
Transfer velocity $v$ [m/s] | 1..2 |  10..20 | 100..200
Homog. B-field $B$ [mT] | 20 | 100 | 1000
Valley splitting $E_{\mathrm{VS,0}}$ [$\mu$eV] | 100 | 200 | 500
Valley $\delta g_{v}/g$ Refs. [64, 73] |  | <10-3 |
Orbital $\delta g_{o}/g$ Refs. [82, 83] |  | <10-3 |
Static QD dephasing $T_{2}^{*}$ [$\mu$s] | 10 | 20 | 50

The detailed simulation of our blueprint SQuS device results in ranges of
orbital energy $E_{\mathrm{orb}}$, QD size $L$ and transfer velocity $v$ given
in Tab. 1. Let us briefly discuss the other parameters from this table. We
consider three different magnetic fields (Tab. 1), the highest of which allows
spin read-out by Zeeman-energy dependent tunneling to a reservoir, while the
others suggest Pauli-spin blockade schemes for spin to charge conversion.
Finally, we use typical variations of effective electron g-factors $\delta
g_{o}$ and $\delta g_{v}$ for orbital and valley state variations,
respectively. The typical $T_{2}^{*}$ time of a quasi-static quantum dot,
depends on the degree of 29Si isotopical purification and might be limited by
the presence of gradient fields as pointed out in Refs. [5, 55, 56] (see
discussion in the next Section). These parameters are used to calculate the
transfer infidelity in the remainder of our paper.

We conclude by pointing out the significance of above-described the modeling
of realistic electrostatic disorder in an optimized device geometry, and its
influence on the properties of the moving QD, on the calculations in the
subsequent Sections. The result that the moving QD does not break apart into
an unintentional DQD motivates the perturbative treatment of orbital and spin
excitations due to dot motion through a disordered channel in Sec. VIB.
Autocorrelation function of electrostatic disorder is calculated in Sec. VI.1
for the optimized structure design (Fig. 7d,f), and using the model of
disorder discussed above. This function provides the key input into the
calculation of the orbital excitation rate in Sec. VI.2 and the motion-induced
spin relaxation rate in Sec. VI.3. Furthermore, the correlation length
obtained from that function determines also the motional narrowing during spin
transfer calculated in Sec. IV. The upper bound for the orbital relaxation
rate in Sec. VID is calculated with respect to the range of $E_{\mathrm{orb}}$
from Fig. 7d. All these rates are then used to estimate the coherent transfer
error due to motion through electrostatically disordered channel in Sec. VIE.
Finally, the variations of the transfer velocity $v$ (Fig. 7f) enter the
discussion of the optimal mean transfer velocity in Sec. VIII.

##  IV Coherent electron spin transfer in Conveyor-Belt without non-adiabatic
effects

In this Section we discuss issues affecting slow operation of the SQuS in the
CB mode, for which the adiabatic character of the charge transfer can be taken
for granted - but spin dephasing mechanisms, leading to finite $T_{2}^{*}$
time for a stationary electron in a QD, have to be considered.

If all the voltages controlling the SQuS are varied slowly enough, the qubit
transfer is be adiabatic, and the electron should remains in the lowest-energy
orbital and valley state while it is being pushed along the channel. We assume
that an external magnetic field is establishing a quantization axis for the
spin, and the only effect of Overhauser field due to nuclear spins of the host
material, spin-orbit interactions, or magnetic field gradients, is to make the
spin splitting of an electron a position-dependent quantity $\hbar\omega(x)$,
where $x$ denotes the position of the QD, understood as the position of the
minimum of the confining potential along the the propagation direction.

A local spin splitting, $\hbar\omega(x)$, has a frozen-in random component,
e.g. due to $g$-factor dependence on the QD position. The influence of such
static disorder in $\hbar\omega(x)$ can be calibrated away: it amounts to a
fixed shift of the phase of the transferred spin superposition. However, the
fluctuations $\hbar\delta\omega(x)$ of $\hbar\omega(x)$ which occur on
timescales shorter than that of accurate measurement of the transferred spin’s
phase, but longer than the timescale of a single qubit transfer, amount to
dephasing of the qubit. This means that the measured coherence of transferred
electron corresponds to a value averaged over a quasi-static distribution of
realizations of $\hbar\delta\omega(x)$. These fluctuations can be caused by
slow nuclear dynamics due to dipolar interaction (for contributions from
Overhauser fields), or low-frequency $1/f$ charge noise leading to slow
changes of electric fields that lead to fluctuations of spin-orbit
interactions affecting the $g$-factor of an electron at given $x$.

If the dot-confined electron is shuttled with velocity $v$, the shuttling
takes $\tau\\!=\\!L_{\mathrm{s}}/v$ and the overall spin phase acquired during
a single transfer will be
$\phi(\tau)=\int_{0}^{\tau}\mathrm{d}t\,\delta\omega(vdt)$. We assume
$\delta\omega(x)$ to be translationally invariant, and that is has a finite
correlation length $l_{c}^{\delta\omega}$, such that the phase variance at the
end of the shuttling process becomes:

$\displaystyle\delta\phi^{2}=\int_{0}^{\tau}\text{d}t\int_{0}^{\tau}$
$\displaystyle\text{d}t^{\prime}\langle\delta\omega(0)\delta\omega(v(t-t^{\prime}))\rangle=$
(2)
$\displaystyle=\delta\omega_{0}^{2}\int_{0}^{\tau}\text{d}t\int_{0}^{\tau}\text{d}t^{\prime}\exp(-\frac{v\absolutevalue{t-t^{\prime}}}{l_{c}^{\delta\omega}})$

where $\langle\ldots\rangle$ denotes the averaging over realizations of
$\delta\omega(x)$, $\delta\omega_{0}^{2}$ is the variance in Larmor precession
frequency of spni in a stationary QD, and we have assumed an exponential decay
of the autocorrelation function of $\delta\omega$. In the expected case of
shuttling distance being larger than the correlation length,
$L_{\mathrm{s}}\gg l_{c}^{\delta\omega}$, we obtain:

$\displaystyle\delta\phi^{2}\big{|}_{L_{\mathrm{s}}\gg l_{c}^{\delta\omega}}$
$\displaystyle\approx\tau\frac{2l_{c}^{\delta\omega}}{v}\delta{\omega}_{0}^{2}=2\left(\frac{\tau}{T_{2}^{*}}\right)^{2}\frac{2l_{c}^{\delta\omega}}{L_{\mathrm{s}}}\,\,.$
(3)

In the above we have identified $\sqrt{2}/\delta\omega_{0}$ with spin
dephasing time $T_{2}^{*}$ observed for stationary spin affected by relevant
sources of quasi-static noise in its spin splitting. The phase variation in
the moving dot is suppressed in comparison to the case of stationary QD for
which $\delta\phi_{v=0}^{2}=2(\tau/T_{2}^{*})^{2}$, and scales linearly with
$\tau$ when $L_{\mathrm{s}}\\!=\\!v\tau$. This is the well-known effect of
motional narrowing of inhomogeneous broadening. If $L_{\mathrm{s}}\gg
l_{c}^{\delta\omega}$ and $v$ is large enough for $\delta\phi^{2}\\!\ll\\!1$,
the loss of spin coherence introduced in Eq. (1) reads

$\delta
C\\!\approx\\!\frac{\delta\phi^{2}}{2}\\!=2\frac{l_{c}^{\delta\omega}L_{\mathrm{s}}}{(vT_{2}^{*})^{2}}\,\,.$
(4)

Making $v$ larger suppresses dephasing - but this dependence will cease to
hold once $v$ becomes too large for the assumption of adiabaticity of electron
transfer to hold.

Let us now use the above formulas to calculate the expected spin dephasing
during an adiabatic evolution using parameters from ranges given in Tab. 1. We
are focusing on transfer across $L_{\mathrm{s}}\\!=\\!10$ $\mu$m with
velocities $v\in[1,200]$ m$/$s. These lead to passage times of
$\tau\\!\in\\![0.05,10]$ $\mu$s.

For Si/SiGe quantum QDs the observed values of $T_{2}^{*}$ are in
$\approx\\!20$ $\mu$s range for isotopically purified silicon. In the presence
of magnetic field gradients this dephasing is caused by charge noise leading
to slow variations in electron’s position along the gradient [5, 6]. For
quasistatic charge noise its correlation length should be limited by
correlation length of static disorder. We use the numerically calculated
static potential disorder from Sec. III.2 to fit the correlation length (see
Fig. 9 in Sec. VIA) to be $l_{c}^{gg}\approx\\!100$ nm. In the worst-case
scenario of magnetic field gradient being constant and equal to value used in
single-dot spin coherence experiments [5, 6], we should thus use
$T_{2}^{*}\approx 20$ $\mu$s and $l_{c}^{\delta\omega}\\!\approx\\!100$ nm. As
shown in Fig. 8 this leads to $\delta C<10^{-3}$ for $v\\!\gtrsim\\!5$ m/s.
With a more realistic assumption that the gradient is sizable only near the
ends of the channel (close to the registers of stationary qubits that need to
be manipulated), taking only 1 $\mu$m as the length of the region in which
charge noise and gradient dominate $T_{2}^{*}$, we get contributions to phase
error that are 10 times smaller, and lead to tolerable phase error in the
whole range of velocities that we consider.

Figure 8: Improvement of spin coherence of an electron transferred
adiabatically along a 10 $\mu$m long channel with its velocity. We plot the
coherence error due to quasistatic noise as a function of velocity for
dephasing times of static spin $T_{2}^{*}\\!=\\!5$, $20$, $100$ $\mu$s, and
two values of spatial autocorrelation length of quasi-static disorder in spin
splitting: $l_{c}^{\delta\omega}\\!=\\!20$ nm (corresponding to typical QD
size) and $l_{c}^{\delta\omega}=100$ nm $\approx l_{c}^{\delta V}$
(corresponding to typical electrostatic disorder autocorrelation length, see
Fig. 9). Lines correspond to Eq. (4) that holds when $\delta\phi\\!\ll\\!1$,
while circles correspond to the formula, $\delta
C\\!=\\!1-e^{-\delta\phi^{2}/2}$, which holds for Gaussian statistics of
random phase fluctuations.

With Si containing 60 ppm of spinful 29Si isotope, the $T_{2}^{*}$ resulting
from interaction with very slowly evolving Overhauser field of the nuclei in
the quantum well is expected to be $\approx\\!10$ $\mu$s [93, 6], and in fact
the nuclear-induced dephasing could be dominated by interaction with a few
29Si and 73Ge nuclei in the barrier for which the predicted $T_{2}^{*}$ could
be as low as $\sim\\!1$ $\mu$s, if the wavefunction overlaps with a few 73Ge
atoms [6]. It is however unclear to what extent these spins are frozen out,
i.e. if the dynamics of Overhauser field generated by them can be considered
ergodic on timescales relevant for quantum computation [6, 94]. Note that
$T_{2}^{*}\\!\approx\\!20$ $\mu$s was observed in Si/SiGe with about 800 ppm
of 29Si [95], consistent with non-ergodic nuclear dynamics. We are however
considering the worst-case scenario of nuclei-induced
$T_{2}^{*}\\!\approx\\!5$ $\mu$s, while making a natural assumption of lack of
spatial correlations between polarization of nuclei, leading to correlation
length given by typical size of the QD,
$l_{c}^{\delta\omega}\\!\approx\\!20\,\mathrm{nm}.$ This gives us the phase
errors $\delta C\\!\leq\\!10^{-3}$ for $v\\!\geq\\!1\,\mathrm{m/s}.$

Finally, in the absence of a gradient, and with even more strongly
isotopically purified samples (or with nuclear dynamics too slow to be
relevant on timescale on which we want to operate our quantum registers), we
are left with a mechanism in which charge noise leads to fluctuations of
electron $g$-factors. We use the parameters for quasi-static g-factor noise
from $T_{2}^{*}$-measurements carried out in MOS devices [60] without magnetic
field gradients. Assuming a Gaussian distribution of quasi-static fluctuations
$g$ factors with rms $\sigma_{g}$, one obtains
$T_{2}^{*}=\sqrt{2}\hbar/\mu_{B}B\sigma_{g}$. The measured values of
$T_{2}^{*}=30\,\mathrm{\mu s}$ at applied magnetic field $B=1.4\,\mathrm{T}$
implies $T_{2}^{*}>100$ $\mu$s in at most a few hundred of mT range that we
consider here. The correlation length is again the QD size,
$l_{c}^{\delta\omega}\\!\approx\\!20$ nm, as the effect of electric fields on
$g$-factor relies to a large degree on presence of atomic length-scale
interface roughness. The phase errors are consequently three orders of
magnitude smaller than the ones given above for the case of Overhauser field
noise, see Fig. 8.

Summarizing, the phase error $\delta C$ due to spatial dependence of the spin
splitting of an electron that is transferred adiabatically is smaller than the
targeted benchmark of $10^{-3}$ for all of the above-discussed values of
$T_{2}^{*}$ and $l_{c}$ when $v\\!>\\!20$ m/s, and for most of them it is
enough for $v$ to be larger than a few m/s, i.e. all the values of $v$ from
the range considered in Tab. 1 are admissible then. The error is the largest
in the case of $T_{2}^{*}\\!=\\!5$ $\mu$s and $l_{c}^{\delta\omega}\approx 20$
nm, relevant for dephasing due to natural concentration of 29Si, or due to
coupling to 73Ge nuclei that are dynamic on the timescale of experiment. In
Fig. 8, we also show the case of even larger error for $T_{2}^{*}\\!=\\!5$
$\mu$s and $l_{c}^{\delta\omega}\approx 100$ nm, relevant for dephasing due to
charge noise in a very large gradient of magnetic field (about 4 times larger
than the ones used in [6, 5]). These cases can be avoided by isotopic
purification and proper design of the gradient magnetic fields.

##  V Spin dephasing due to nonadiabaticity of electron dynamics at larger
velocities

We now move to the regime of larger $v$, in which non-adiabatic character of
the charge transfer has to be seriously considered. We will discuss how the
presence of orbital/valley level dependent spin precession together with not-
fully adiabatic character of charge transfer, open up a highly relevant
channel of spin dephasing.

The adiabaticity of the SQuS operation in CB mode, i.e. lack of excitation of
the electron out of its instantaneous lowest energy state $\ket{g_{o,v}}$, is
not necessary for the charge transfer to occur: the electron will traverse the
channel unless the excitations cause the electron to become “lost” by being
excited from the pre-existing/moving QDs into the continuum of high-energy
states, or by being trapped in an unintentional deep potential well. Device
optimization presented in Section III makes the probability of such events
low. However, even without them, the non-adiabaticity leads to randomness in
state trajectory. Transitions between ground and excited orbital states of the
moving QD happen due to electrostatic disorder (Fig. 3) that acts as a time-
dependent perturbation of the confinement potential in the frame of reference
moving with the electron-confining QD. Furthermore, when the wavefunction of
the electron confined in the moving QD overlaps with defects in the interface
such as atomistic steps or interface gradients, orbit-valley coupling is
activated [68, 1, 96, 97], and its time-dependence excites the electron into
the higher-energy valley state.

These processes lead to finite probability of excitation into a higher energy
orbital/valley state. After such an excitation, relaxation back to the ground
state will occur on timescale of $\tau_{r}$. When this $\tau_{r}$ is smaller
than the total shuttling time $\tau$, or when the excitation occurs at a
random place along the channel, the electron will spend a randomly distributed
time in an excited state. Spin precession frequency $\omega$ is expected to
depend on the orbital/valley state. Random time spent in a state characterized
by random $\omega$ results in random contribution to spin phase characterized
by rms $\delta\phi$. For a probability of excitation given by $p_{e}$, the
phase error is then given by Eq. (1). A more detailed formal derivation of
that formula is given in Appendix B.

Let us discuss the physical mechanisms leading to state-dependent precession
rates. The first mechanism is the state-dependence of the effective $g$-factor
of the electron: while $g\\!\approx\\!2$ in silicon, its value is exhibiting
relative variation of $\delta g_{o}/g\\!\lesssim\\!10^{-3}$ when we compare
electrons in states localized in two distinct QDs, or ground and excited
states of a single QD, and similarly $\delta g_{v}/g\\!\lesssim\\!10^{-3}$ in
each of the two valley states [64, 73]. This g-factor variation is caused by
finite spin-orbit coupling and electrostatic/interface disorder [81, 66]. If
the electron spends the time $\tau_{e}$ in an excited state, the additional
spin phase acquired by it compared to the case of adiabatic evolution is

$\delta\phi_{g}\approx 2\frac{\delta
g}{g}\frac{\mu_{B}B\tau_{e}}{\hbar}\approx 350\frac{\delta
g}{g}B[\mathrm{T}]\tau_{e}[\mathrm{ns}]\,\,.$ (5)

For the typical value of $\delta g/g\approx 10^{-3}$, this yields
$\delta\phi_{g}\\!\approx\\!0.35B[\mathrm{T}]\tau_{e}[\mathrm{ns}]$. Note that
$\tau_{e}$ is given by the relaxation time $\tau_{r}$ if
$\tau_{r}\\!\ll\\!\tau$, and it is $\tau_{e}\\!\lesssim\\!\tau$ when
relaxation is too slow, and an electron excited at some position along the
channel will stay in the higher-energy state for the rest of the shuttling
time.

The second mechanism is the state-dependence of spin-orbit coupling for the
two different valley-states. An analogous mechanism for orbital excitation is
neglected here, because orbital excitations relax quickly via phonon emission,
$\tau_{e}\\!=\\!\tau_{r}\\!<\\!1$ ns for the $E_{\mathbf{orb}}$ range found in
Fig. 7d, as will be discussed Sec. VI. After a Galilean boost of the spin-
orbit interaction Hamiltonian into the frame co-moving with the QD, we obtain

$H_{\mathrm{SO,i_{v}}}=\alpha_{+,i_{v}}m^{*}v_{[1\overline{1}0]}\sigma_{[110]}+\alpha_{-,i_{v}}m^{*}v_{[110]}\sigma_{[1\overline{1}0]},$
(6)

where $i_{v}\in\\{v_{+},v_{-}\\}$ labels the lowest-energy valley states,
$\alpha_{\pm,i_{v}}=(\alpha_{i_{v}}\pm\beta_{i_{v}})$ is the valley dependent
spin-orbit strength due to Rashba and (effective) Dresselhaus 2D effects in
the Si/SiGe system, and $v_{[1\overline{1}0]}$ and $v_{[110]}$ are the
instantaneous QD velocities along along the $[1\overline{1}0]$ and $[110]$
crystal axes. The first order correction to the valley-dependent spin
precession frequency is then

$\delta\omega_{\mathrm{SO},i_{v}}=\frac{m^{*}}{\hbar}\left(\alpha_{+,i_{v}}v_{[1\overline{1}0]}\sin(\phi_{\mathrm{B}})+\alpha_{-,i_{v}}v_{[110]}\cos(\phi_{\mathrm{B}})\right),$
(7)

with $\phi_{\mathrm{B}}$ being the angle between the ${[1\overline{1}0]}$
crystal axis and the external magnetic field $\bm{B}$. A suitable choice of
electron velocity $\bm{v}$ and magnetic field direction parameterized by
$\phi_{\mathrm{B}}$ may reduce the impact of this effect. In particular,
choosing $\bm{v}\parallel\bm{B}$ (i.e. $\phi_{\mathrm{B}}=0$) being oriented
along either $[1\overline{1}0]$ or $[110]$ eliminates the above first order
contribution. However, to realize a two dimensional spin qubit architecture,
binding the shuttling direction to the external magnetic field orientation is
undesirable.

Assuming that the spin-orbit coefficients are simply opposite in sign in
different valleys [84], and using an estimate from SiMOS device
$|\alpha|\\!\approx\\!50$ nm/ns [66], we find

$\delta\phi_{\mathrm{SO}}\approx 1.6\cdot
10^{-4}v\left[\frac{\mathrm{nm}}{\mathrm{ns}}\right]\tau_{e}[\mathrm{ns}]\,\,.$
(8)

Here, $|\alpha|$ has been introduced as the general strength of spin-orbit
coupling. As the spin-orbit interaction is expected to be weaker in Si/SiGe
nanostructure [56], the above gives an upper bound on the spin-orbit induced
phase rotation in the considered SQuS. The second-order contributions to
precession frequency due to transverse fields will give corrections of the
order of $|\alpha|m^{*}v/E_{Z}\\!\ll\\!1$ to this formula.

We have singled out the secular spin-valley interaction term (of type
$\sigma_{z}\tau_{z}$), since both spin and valley precession frequencies – as
well as their difference – are large compared to the strength of the
interaction ($E_{\mathrm{Z}}/h>0.1\,\mathrm{GHz}$ and $E_{\mathrm{VS}}\gg
E_{\mathrm{Z}}$, while $|\alpha|m^{*}v/h\leq 1\,\mathrm{MHz}$), justifying a
secular approximation. An estimate for the error due to the transverse
components is the misalignment of the spin quantization axes in the two valley
subspaces. Again assuming spin-orbit coefficients of opposite signs, this
amounts to $(|\alpha|m^{*}v/E_{\mathrm{Z}})^{2}<10^{-4}$ for magnetic fields
larger than $5\,\mathrm{mT}$ and shuttling speeds ($v<100\,\mathrm{m/s}$).
There is the remaining effect that spatial variations in spin-orbit strengths
are rendered dynamical due to the motion of the QD. As shown in [33], spin
relaxation time due to such effectively dynamical transverse spin-orbit fields
is $T_{1}\\!>\\!1$ ms for $v\\!\leq\\!100$ m/s. This corresponds to bit flip
error $<10^{-4}$ for a qubit moved by $10$ $\mu$m (see alse section VIC).

Before moving to calculations of $p_{e}$ and $\tau_{r}$ in the subsequent
sections, let us check in which parameter ranges we expect the phase error
$\delta C$ to be below our targeted threshold of $10^{-3}$. Since $\delta
C\\!\leq\\!p_{e}$ according to Eq. (1), when $p_{e}\\!<\\!10^{-3}$, the phase
error is guaranteed to be below the targeted threshold no matter what
$\tau_{e}$, and thus $\delta\phi$, is. Only for $p_{e}\\!>\\!10^{-3}$ we have
to rely on $\delta\phi\\!\ll\\!1$ to find ourselves in the regime in which
$\delta C\\!\approx\\!p_{e}\delta\phi^{2}/2\\!<\\!10^{-3}.$ The spin-orbit
phase contributions from Eq. (8) are $\delta\phi_{\mathrm{SO}}<1$ when
$v[\mathrm{m/s}]\tau_{e}[\mathrm{ns}]\\!<\\!10^{4}$, so even for
$v\\!\approx\\!100$ m/s, relaxation times below 100 ns make $\delta
C\\!\ll\\!p_{e}$. On the other hand, having no relaxation during shuttling
time $\tau$ makes $\delta C\\!\approx\\!p_{e}$, as in that case
$\tau_{e}\\!\approx\\!\tau$ and $\delta\phi_{\mathrm{SO}}\\!\approx\\!1.6$.
For phase variations due to randomness in $g$-factors, from Eq. (5)with
$\delta g/g\\!=\\!10^{-3}$, we have $\delta\phi_{g}\\!<\\!1$, if
$\tau_{e}\\!<\\!140\,$ ns at $B\\!=\\!20$ mT ($\tau_{e}\\!<\\!14\,$ ns at
$B\\!=\\!200$ mT). Note that when $\tau_{e}\\!\approx\tau$, we have
$\delta\phi_{g}\\!\approx\\!3.5\cdot(10^{1}-10^{3})B[T]$ for $v\in[1,100]$ m/s
range. In this case, $\delta\phi_{g}\\!<\\!1$ only for lowest magnetic fields
of $\approx 20$ mT compatible with ESR control of spin qubits, and for the
highest considered $v\\!\approx\\!100$ m/s (corresponding to
$\tau\\!\approx\\!100$ ns). Note that in this case the electron will finally
have to undergo relaxation to its ground valley state at its final
destination, and the stochastic nature of this relaxation will lead to
additional dephasing.

Clearly, working at lowest $B$ fields compatible with qubit control is
beneficial if we cannot suppress orbital/valley excitations so that $p_{e}$ is
below the error threshold: having $B\\!\approx\\!20$ mT strengthens the phase
error suppression when $\tau_{r}\\!\ll\\!100$ ns, and it is necessary for any
such suppression when $\tau_{r}\\!>\\!\tau\approx 100$ ns at the highest
considered $v$.

##  VI Coherent electron transfer in Conveyor-Belt in presence of orbital non-
adiabaticity and motion-induced spin relaxation

Movement of the quantum dot turns the spatial electrostatic disorder into
dynamical charge noise in the frame co-moving with the dot. This noise can
then cause transitions between orbital states, and it can also couple to spin
degree of freedom by spin-orbit interaction. The latter mechanism was first
considered in [33]. Here we use numerical simulations of electrostatic
disorder from Sec. III to estimate the orbital excitation rate. Then we
calculate the orbital relaxation rate due to phonon emission, and combine both
orbital transition rates to estimate the spin dephasing caused by mechanism
discussed in previous Section. Furthermore, using the autocorrelation function
of disorder calculated here as an input into the theory from [33], we prove
that spin relaxation along the channel is not an obstacle for reaching the
targeted qubit transfer fidelities.

###  VI.1 Autocorrelation function of electrostatic disorder

We start by determining the correlation length $l_{c}$, and variance $\delta
V^{2}$, of the potential disorder along the channel for the optimized geometry
(Fig. 7d,f). We calculate the autocorrelation function of the matrix element
$M_{\text{gg}}(x_{0})=\bra{g_{o}}\delta V(x)\ket{g_{o}}$, which quantifies the
effects of the electrostatic disorder on the electron occupying the ground
state of the QD localized at $x_{0}$. We compute this function, defined as

$\displaystyle K_{\text{gg}}(x_{0},x_{0}+x)$ $\displaystyle=\langle
M_{\text{gg}}(x_{0})M_{\text{gg}}(x_{0}+x)\rangle$ $\displaystyle-\langle
M_{\text{gg}}(x_{0})\rangle\langle M_{\text{gg}}(x_{0}+x)\rangle$ (9)

by fixing $x_{0}$ in a simulated device having length of 10 SQuS unit cells,
and averaging over 80 realizations of disorder. To create an effectively 1D
potential we additionally integrate over the y axis, i.e.
$M_{\text{gg}}(x_{0})=\int|\psi_{g}(x^{\prime}-x_{0},y^{\prime})|^{2}V(x^{\prime},y^{\prime})\text{d}y^{\prime}\text{d}x^{\prime}$.
At each position $x_{0}$ of the QD, and for each disorder realization, we
solve the Schrödinger equation and fit ground-state wavefunctions (Gausssian
shapes with variable width) to the ansatz
$\psi_{g}(x^{\prime},y^{\prime})=\psi_{g}^{(x)}(x^{\prime})\cdot\psi_{g}^{(y)}(y^{\prime})$
in both directions x and y.

We perform this calculation for 250 values of $x_{0}$, and as shown in Fig. 9,
$K_{\text{gg}}(x,x_{0})$ is only slightly dependent on the choice of $x_{0}$,
which confirms its approximate stationarity, i.e.
$K_{\text{gg}}(x_{0},x_{0}+x)\approx K_{\text{gg}}(x)$, and justifies further
averaging of the results over $x_{0}$ (solid green line). For small $x$, the
correlation function flattens due to finite size of electron wavefunction
$\psi_{g}(x)$, which effectively filters out the disorder fluctuations on
lengthscales below the QD size, $L_{\text{dot}}$. An exponential fit of
$x\\!>\\!L_{\text{dot}}$ of the
$K_{\text{gg}}(L_{\text{dot}})e^{-|x-L_{\text{dot}}|/l_{c}^{gg}}$ form, gives
an estimate of the correlation length $l_{c}^{gg}\\!=\\!90$ nm while the
standard deviation of the disorder-induced energy shift of the QD ground state
is $\sqrt{\delta V^{2}}\approx\sqrt{K_{\text{gg}}(0)}\\!=\\!2.8$ meV. Note
that $l_{c}^{gg}\\!>\\!L_{\text{dot}}$ confirms that the disorder in the
optimized design does not vary fast enough to break the QD apart, in agreement
with other calculations from Sec. III. The good fit at
$x\\!>\\!L_{\text{dot}}$ means that the correlation length of the matrix
element $M_{\text{gg}}$ is a good indicator of the correlation length of the
1D potential, i.e. we can approximate the autocorrelation of $\delta V(x)$
itself, $K_{\delta V}(x)$, by an exponential with correlation length
$l_{c}^{\delta V}\\!\approx\\!l_{c}^{gg}\gg L_{\text{dot}}$.

Figure 9: Autocorrelation function $K_{\text{gg}}(x_{0},x_{0}+x)$ of the
matrix element $\delta M_{\text{gg}}(x_{0})=\bra{g_{x_{0}}}\delta
V(x)\ket{g_{x_{0}}}$, defined in Eq. (VI.1), calculated for device geometry
and voltages used in Fig. 7d,f, and averaged over 80 disorder realizations.
Blue lines correspond to 250 distinct values of $x_{0}$. We plot averaged
autocorrelation (solid green line) and an exponential fit (dashed red line)
for $x\\!>\\!L_{\text{dot}}$ of the
$K_{\text{gg}}(L_{\text{dot}})e^{-\absolutevalue{x-L_{\text{dot}}}/l_{c}^{gg}}$
form. The fitted values $\sqrt{K_{\text{gg}}(0)}\\!=\\!2.8$ meV and
$l_{c}^{gg}\\!=\\!90$ nm indicate typical amplitude and correlation length of
effective 1D disorder $\delta V(x)$. The long correlation length
$l_{c}^{\delta V}\gg L_{\text{dot}}$ is an indicator of stability of the
moving QD against potential disorder.

###  VI.2 Orbital excitations due to electrostatic disorder

Motion of the QD converts the electrostatic disorder into a time-dependent
electric field in the frame of the QD. Due to $1/f$ character of charge noise,
electrostatic disorder can be treated as static during a single realization of
electron transfer, but it varies between consecutive realizations of the
shuttling protocol, and thus in order to calculate the probability of orbital
excitation of the electron, one should average over realizations of
electrostatic disorder, i.e. average over realizations of electric noise
experienced by the electron confined in the moving QD. This electric noise
will cause transitions from ground to excited orbital states, if it has
appreciable spectral power at the frequency close the $E_{\text{orb}}/\hbar$.
Such transitions are not captured by solving the time-independent Poisson-
Schrödinger equation in Sec. III. There, only the $E_{\text{orb}}$ and
velocity variations of the moving QD have been calculated (Fig. 7d,f).

In a single realisation, the electron,treated here as a two-level system, with
Hilbert space spanned by ground and excited orbital states, effectively feels
a transverse time-dependent field, which in the frame of reference of the
moving QD is defined as:

$\delta M_{\text{eg}}(x_{0})=\bra{e_{o}}\delta V(x+x_{0})\ket{g_{o}},$ (10)

where $x_{0}=vt$, $\delta V(x)$ is a random contribution to the effective 1D
potential and $\ket{g_{o}},\ket{e_{o}}$ are ground and first excited orbital
state of the moving harmonic potential, respectively. We calculate the
excitation probability in the first order of perturbation theory by averaging
over the realizations of quasistatic noise in $\delta V(x)$, which translates
into averaging over realizations of $\delta M_{\text{eg}}(x_{0})$. We follow
Refs. [98, 99] and relate the excitation rate with the spectral density of the
dynamical noise at the frequency corresponding to the gap, i.e.

$\Gamma_{+,o}=\frac{1}{\hbar^{2}}\int\text{d}(\Delta t)\langle\delta
M_{\text{eg}}(v\Delta t)\delta M_{\text{eg}}(0)\rangle
e^{-iE_{\text{orb}}\Delta t/\hbar},$ (11)

where $\langle\ldots\rangle$ denotes averaging over realizations of
electrostatic disorder $\delta V(x)$. Using the autocorrelation of $\delta
V(x)$ of the form $K_{\delta V}=\delta V^{2}e^{-|x|/l_{c}^{\delta V}}$, one
obtains (see Appendix. C for details) a result that holds in the regime of
$v\\!\ll\\!E_{\mathrm{orb}}l_{c}^{\delta V}/\hbar\\!\sim\\!10^{5}$ m/s:

$\Gamma_{+,o}=\frac{\delta
V^{2}}{\hbar^{2}v}\frac{L_{\text{dot}}^{2}}{l_{c}^{\delta
V}}\exp(-\frac{1}{2}\Big{[}\frac{E_{\text{orb}}L_{\text{dot}}}{\hbar
v}\Big{]}^{2}).$ (12)

We can see that for $v/L_{\text{dot}}\ll E_{\text{orb}}/\hbar$
($v\\!\ll\\!10^{4}$ m/s) the rate is suppressed by a Gaussian factor.

###  VI.3 Spin relaxation

An analogous calculation can be (and in fact was, [33]) performed for
transitions between two spin states of the electron due to spin-orbit coupling
being time-dependent in the frame co-moving with the QD across an
electrostatic disorder, with the latter modulating the local value of spin-
orbit coupling term. As calculated in [33], using the model of the
electrostatic disorder with exponentially decaying correlations that we have
employed above, the spin-relaxation rate is given by:

$\Gamma_{-,\text{spin}}=\left(\frac{2\hbar\delta V}{l_{c}^{\delta
V}E_{\text{orb}}^{2}}\right)^{2}\left(\frac{v\,\omega^{2}l_{c}^{\delta
V}}{v^{2}+(\omega l_{c}^{\delta V})^{2}}\right)\alpha^{2},$ (13)

where $\alpha$ is the spin-orbit coupling and $\hbar\omega$ is the Zeeman
splitting. Using the above, the estimated probability of spin flip after the
transfer follows from $\delta p_{\uparrow}=\Gamma_{-,\text{spin}}L_{s}/v$, and
it is largest in the limit of small velocities, $v\ll\omega l_{c}^{\delta
V}/\hbar$. We obtain then

$\delta p_{\uparrow}\leqslant 4\hbar^{2}\frac{\delta
V^{2}L_{s}}{(l_{c}^{\delta
V})^{3}E_{\text{orb}}^{4}}\alpha^{2}\approx\frac{4\times
10^{-4}}{(E_{\text{orb}}[\text{meV}])^{4}}\,\,,$ (14)

for parameters obtained from the numerical simulations of electrostatic
disorder, i.e. for $\delta V=3\,\mathrm{meV}$ and $l_{c}^{\delta V}=100$ nm,
and using $\alpha\\!=\\!50$ m$/$s. Since the latter is almost certainly a
generous overestimate for Si/SiGe, as discussed previously in Sec. (V), and we
are targeting the orbital excitation gaps $E_{\mathrm{orb}}\\!\geq\\!1$ meV,
this result shows that effects of spin relaxation due to electrostatic
disorder and spin-orbit coupling in CB are not endangering the goal of keeping
the error rate below $10^{-3}$ threshold. In Fig. 1 we have shown the result
from this Equation for $\alpha\\!=\\!5$ m$/$s taken from [33] and
$E_{\mathrm{orb}}\\!=\\!1$ meV as a dashed line.

###  VI.4 Orbital relaxation due to electron-phonon coupling

As the relevant relaxation mechanism, we only consider phonon mediated
relaxation. Other relaxation mechanisms may dominate at low orbital energies
in stationary QDs, located close to charge fluctuators such as electron
reservoirs [56], but we neglect these effects when considering a shuttled QD
far away from such regions, and assume that high frequency noise on the gates
may be sufficiently suppressed in the experiment.

To compute the orbital relaxation due to phonons, we employ Fermi’s Golden
Rule in the zero-temperature limit [100, 97], as
$k_{\mathrm{B}}T\\!\ll\\!E_{\mathrm{orb}}$ for
$T\\!\approx\\!20\,\mathrm{mK}$, so that temperature enters only as slight
modification factor in the relaxation rate. The relevant orbital-phonon
coupling will only be due to deformation potential, in contrast to GaAs
heterostructures where also piezoelectric coupling is present [87, 101].
Employing the Herring-Vogt deformation potential Hamiltonian to model this
interaction [100], and taking into account that only the $[001]$-valleys are
occupied due to the strain in the QW, we arrive at a relaxation rate of

$\frac{1}{\tau_{r,\text{o}}}=\frac{E_{\mathrm{orb}}^{3}}{8\pi^{2}\hbar^{4}\rho}\left(\frac{\Xi_{\mathrm{d}}^{2}I_{0}+2\Xi_{\mathrm{d}}\Xi_{\mathrm{u}}I_{2}+\Xi_{\mathrm{u}}^{2}I_{4}}{v_{l}^{5}}+\frac{\Xi_{\mathrm{u}}^{2}J}{v_{t}^{5}}\right)$
(15)

written in terms of dilatation and shear deformation potentials $\Xi_{d}$,
$\Xi_{u}$, transverse and longitudinal speeds of sound $v_{t}$ and $v_{l}$,
and the mass density of silicon $\rho$, respectively [97]. The relevant
integrals are given by

$I_{n}=\int_{0}^{2\pi}\mathrm{d}\phi_{k}\int_{0}^{2\pi}\mathrm{d}\theta_{k}\sin(\theta_{k})\cos^{n}(\theta_{k})|\langle
g_{o}|e^{i\bm{k}_{t}\cdot\bm{\hat{r}}}|e_{o}\rangle|^{2}$ (16)
$J=\int_{0}^{2\pi}\mathrm{d}\phi_{k}\int_{0}^{2\pi}\mathrm{d}\theta_{k}\sin^{3}(\theta_{k})\cos^{2}(\theta_{k})|\langle
g_{o}|e^{i\bm{k}_{l}\cdot\bm{\hat{r}}}|e_{o}\rangle|^{2}$ (17)

with $\bm{k}$ being the angular wave vector of the phonons matched to the
orbital energy splitting $\bm{k}=[E_{\mathrm{orb}}/(\hbar
v_{\lambda})]\bm{e}_{\bm{k}}\equiv k_{0}\bm{e}_{\bm{k}}$ with
$\bm{e}_{\bm{k}}$ the unit vector along the direction of $\bm{k}$.
$|g_{o}\rangle$ and $|e_{o}\rangle$ are the ground and first excited orbital
state, respectively. Making a harmonic confinement ansatz for the three
spatial directions with length scales $L_{x}$, $L_{y}$ and $L_{z}$ allows us
to perform the $\phi_{k}$ integration analytically:

$\displaystyle\int_{0}^{2\pi}$ $\displaystyle\mathrm{d}\phi_{k}|\langle
g_{o}|e^{i\bm{k}_{\lambda}\cdot\bm{\hat{r}}}|e_{o}\rangle|^{2}=2\pi(k_{0}L_{\text{dot}})^{2}\sin^{2}(\theta_{k})$
$\displaystyle\times\exp(-\frac{k_{0}^{2}}{2}\big{[}L_{z}^{2}\cos^{2}(\theta_{k})+L_{\text{dot}}^{2}\sin^{2}(\theta_{k})\big{]}).$
(18)

In the above, we assumed that the in-plane confinement is roughly isotropic
($L_{x}\approx L_{y}=L_{\text{dot}}$), which is a sensible approximation for
the considered QD shapes, cf. Fig. 6b and 7d. The $\theta_{k}$ integration is
then carried out numerically and the results are plotted in Fig. 10. Most
importantly, the orbital relaxation time $\tau_{r,o}$ is shorter than 100 ps
for the considered orbital energies from 1 to 3 meV (Tab. 1). Such an
confinement can be achieved for the moving QD across the SQuS according to
Fig. 7d. Thus, orbital relaxation is sufficiently fast to relax back to the
orbital ground-state before a significant difference in spin-phase can be
accumulated in the moving QD.

Figure 10: Orbital relaxation time in a circular QD ($L_{x}=L_{y}$), with
$L_{z}$ as the remaining parameter. $L_{z}\rightarrow 0$ illustrates the
fictious case of a completely 2D electron (where phonon-bottlenecking along
the $z$-direction is absent), while $L_{z}=4\,\mathrm{nm}$ is a reasonable
upper bound for the localisation of the electron wavefuntion in z-direction.

A single QD exhibits strong orbital relaxation, attributed to the large
orbital splitting $\sim\\!1\,\mathrm{meV}$, matching a large phonon density of
states at these frequencies, and the strong orbital-phonon coupling, which
allows phonons to siphon excess orbital energy on below $100\,\mathrm{ps}$
timescales. As phonons in Si are relatively slow, and therefore have short
wavelengths approaching the characteristic QD dimension at the relevant
frequencies, bottlenecking effects where orbital-phonon coupling becomes
inefficient due to matching/exceeding of phonon wavelengths to the QD size
(suppression of the coupling elements $\langle
g_{o}|e^{i\bm{k}\cdot\bm{\hat{r}}}|e_{o}\rangle$) become relevant. However, as
the characteristic QD size scales with $1/\sqrt{E_{\mathrm{orb}}}$, the
relevant parameter is $k_{0}L_{\mathrm{dot}}\propto\sqrt{E_{\mathrm{orb}}}$.
This is sufficient to soften the effect of bottlenecking in the relevant
energy regime $1-3\,\mathrm{meV}$.

While the effect of phonon bottlenecking is of significance at characteristic
orbital energy scales, its effects are reduced in two ways. Firstly, the small
parameter for bottlenecking for the in-plane components $x$ and $y$ is
$k_{0}L_{x}=\sqrt{E_{\mathrm{orb}}/(m^{*}v_{\lambda}^{2})}$ and only takes on
moderate values ($\approx 5$ at $E_{\mathrm{orb}}=1\,\mathrm{meV}$) for the
relevant energies, as discussed above. Secondly, the much stronger confinement
along the growth direction ($z$-direction) leads to an additional suppression
of these effects ($k_{0}L_{z}=\frac{E_{\mathrm{orb}}}{\hbar
v_{\lambda}}L_{z}$), which is $\approx 1$ for $L_{z}\approx 4\,\mathrm{nm}$
and $E_{\mathrm{orb}}=1\,\mathrm{meV}$.

###  VI.5 Transfer infidelity due to orbital non-adiabaticity

Let us first estimate the final occupation of the excited orbital state
$p_{e,o}$ for the typical case, in which the shuttling time is much longer
that the relaxation time $\tau\\!\gg\\!\tau_{r}$. We can then estimate
$\overline{p}_{\text{e,o}}\\!\approx\\!\Gamma_{+,o}\tau_{\text{r,o}}$ from the
steady-state value. For typical $\tau_{r,o}\sim 10$ ps, the final occupation
of the excited state is always below $\overline{p}_{e,o}\leqslant 10^{-3}$ as
long as $\Gamma_{+}\leq 0.1\,\mathrm{ns}^{-1}$. In our design, the excitation
rate due to static disorder computed in Eq. (12) gives non-negligible
$\Gamma_{+}\sim 10^{-4}\mathrm{ns}^{-1}$ only at $v\sim 4000$ m/s
($E_{\text{orb}}=1\,\mathrm{meV}$, $\sqrt{\langle\delta V^{2}\rangle}\\!=\\!3$
meV and $l_{c}^{\delta V}=100$ nm), and it is suppressed by a Gaussian factor
at lower $v$.

Now, let us estimate the spin dephasing caused by repeated processes of
orbital excitation followed by relaxation that occur during the shuttling.
According to the model used in Eq. (5), for time spent in the excited state
given by typical orbital relaxation time, $\tau_{e,o}\approx 10\,\mathrm{ps}$,
the variance of random phase acquired after each relaxation event is given by:

$\delta\phi_{g}^{2}\approx(\delta\omega\,\tau_{e,o})^{2}\approx
10^{-4}\big{(}\hbar\delta\omega[\mu\text{eV}]\big{)}^{2},$ (19)

where $\delta\omega=\delta g_{o}\mu_{B}B/\hbar$ is the difference of Larmor
frequencies between ground and excited orbital state. As the relaxation is
expected to be orders of magnitude faster than the excitation, we estimate
total coherence error as the phase error per relaxation event
$\delta\phi_{g}^{2}$ /2 times the number of transitions from ground to excited
state, which depends on the excitation rate and the shuttling time
$N_{e}\\!\approx\\!\Gamma_{+,o}\tau$. In this way we estimate total phase
error due to temporal occupation of excited orbital state during the shuttling
as

$\delta
C\approx\Gamma_{+,o}\,\tau\,\delta\phi_{g}^{2}/2=\frac{(\hbar\delta\omega[\mu\text{eV}])^{2}\Gamma_{+}[\text{ns}^{-1}]}{2v[m/s]},$
(20)

which can be used to define a tolerable level of excitation rate. Using
parameters from Tab. 1, in the non-optimal (for spin coherence) regime of
large magnetic field $B=1$ T, g-factor difference $\delta g_{o}/g=10^{-3}$ and
velocity of $v=10$ m/s (transfer time $\tau\sim 1\,\mu\mathrm{s}$), the phase
error below the threshold $\delta C\\!=\\!10^{-3}$ requires the excitation
rate $\Gamma_{+,o}\\!<\\!1\,[\mathrm{ns}^{-1}]$. This value is orders of
magnitude larger than the above-estimated transition rate due to electrostatic
disorder. Even if sources of orbital excitations other than charge disorder
simulated in Sec. III, e.g. charged defects in the channel or threading
dislocations, are relevant, the excitation rate associated with them would
have to be $\gtrsim 1\,\mathrm{ns}^{-1}$ for dephasing to become dangerous due
to orbtial excitation.

We hence come to the important conclusion that spin dephasing due to orbital
non-adiabatic effects (and also qubit state error due to motion-induced spin
relaxtion), should not pose a limitation for coherent electron transfer in the
CB mode. See Figs. 1 and 15 for the comparison to other, more relevant
mechanisms considered throughout the paper.

##  VII Coherent electron transfer in Conveyor-Belt in presence of valley
degree of freedom

After analyzing dephasing due to quasistatic noise in Sec. IV and due to
temporal occupation of higher orbital state in Sec. VI, we finally analyze the
phase error resulting from non-adiabatic evolution of the valley degree of
freedom.

###  VII.1 Model of instantaneous valley states

Figure 11: Modelling of the local valley splitting across the SQuS channel.
(a) A 2D cross-section through a Si/SiGe interface region with a gate definied
quantum dot containing the electron. A single atomistic step is indicated at
the center, with Ge-alloy discorder smearing out the location of the
interface. Grey dots represent Si atoms in the strained Si-layer and QW
barrier region, respectively. Dark green dots represent Ge atoms. The angle
$\vartheta$ indicates the average effective misscut. (b) A 1D linecut along
the growth direction which is the model containing all short-range disorder
discussed in the main text. The models describing long range disorder in the
2D-plane perpendicular to the growth-direction are sketched in (c) (step
model) and (d) (gradient model), with the color coding indicating the
continuous and discrete spatial variation in valley phase respectively. The
angle $\theta$ in (c) is the angle between step normals and shuttling
direction.

The relevant degree of freedom with low energy splitting in the Si/SiGe
quantum wells discussed in this paper are the conduction band minima (valleys)
along the growth direction in $k$ space (labelled as $|k_{[001]}\rangle$ and
$|k_{[00\overline{1}]}\rangle$), with the in-plane valleys
($|k_{[100]}\rangle$, $|k_{[\overline{1}00]}\rangle$, $|k_{[010]}\rangle$ and
$|k_{[0\overline{1}0]}\rangle$) being much higher in energy due to strain [1].
In gate-defined Si/SiGe QDs, reported values of the splitting between the
valleys $|k_{[001]}\rangle$ and $|k_{[00\overline{1}]}\rangle$ vary between
$10$ and $\approx 200$ $\mu$eV [69, 70, 64, 71, 72, 28, 13, 73, 74, 55, 56].
It can exceed 500 $\mu$eV in MOS structures [52, 54, 75, 76]. Crucially, the
valley splitting $E_{\text{VS}}$ in Si/SiGe is a local property of the
heterostructure depending on atomic steps and Ge segregation at the Si/SiGe
interface [102, 78]. Variations of electric field and crystal compositions
that are spatially smooth on the scale of the lattice constant affect the
local value of $E_{\text{VS}}$, but they do not lead to valley-orbit mixing
that would couple the valley degree of freedom with the electron motion.
Perturbations on a length scale of the Si lattice spacing e.g. atomic steps at
interface of the quantum well [103], Ge segregation and SiGe alloy
disorder[102] (see Fig. 11 (a)), however, severely affect not only
$E_{\text{VS}}$, but also the composition of valley eigenstates [67, 77, 68,
56, 78, 79]. We parameterize the inhomogeneous valley splitting
$E_{\text{VS}}$ using an effective valley parameter termed bare valley
splitting $E_{\text{VS,0}}$, which is assumed to be approximately homogeneous
on a length scale of the SQuS device. This $E_{\text{VS,0}}$ is the expected
valley splitting in absence of the models of interface perturbations discussed
below (interface step and smooth interface gradient) and therefore the
theoretical maximum value of $E_{\text{VS}}$ observed in a QD located
somewhere along the channel. It includes the influence of alloy disorder, the
dependence on the electric field along the QW confinement direction, the Ge
content of the barrier, the thickness of the QW layer and the mean length-
scale of Ge segregation at the interface.

While a complete description of atomistic interface disorder would require 3D
modelling, we restrict ourselves to a 2D model as will be specified below [77,
104, 96, 97]. As the orbital and valley degrees of freedom are only weakly
coupled, we may, to a good approximation, find an effective valley Hamiltonian
by averaging over the two-dimensional spatial probability density $\rho(x,y)$.
Introducing the valley operators $\hat{\tau}_{x}=|k_{[001]}\rangle\langle
k_{[00\overline{1}]}|+\mathrm{h.c.},$
$\hat{\tau}_{y}=-i|k_{[001]}\rangle\langle
k_{[00\overline{1}]}|+\mathrm{h.c.},$ we can write the averaged Hamiltonian as

$\displaystyle\hat{H}_{\mathrm{v}}(x_{0})=\frac{E_{\mathrm{VS,0}}}{2}\iint$
$\displaystyle\rho(x-x_{0},y)\left[\cos\left(\varphi_{\mathrm{VS}}(x,y)\right)\hat{\tau}_{x}\right.$
$\displaystyle\left.+\sin\left(\varphi_{\mathrm{VS}}(x,y)\right)\hat{\tau}_{y}\right]\mathrm{d}x\,\mathrm{d}y,$
(21)

where $\rho(x-x_{0},y)$ is the electron probability density centered around
$(x_{0},y_{0})$, $y_{0}$ represents a constant position in the direction
perpendicular to shuttling, and $\varphi_{\mathrm{VS}}(x,y)$ is the spatially
dependent valley-field. We will consider two models of
$\varphi_{\mathrm{VS}}(x,y)$, describing the extreme cases of its gradual and
instantaneous change. Both model can be expressed in terms of an effective
Hamiltonian,

$\hat{H}_{\mathrm{v}}(x_{0})=\frac{E_{\text{VS}}(x_{0})}{2}\Big{[}\cos(\tilde{\varphi}_{\text{VS}}(x_{0}))\hat{\tau}_{x}+\sin(\tilde{\varphi}_{\text{VS}}(x_{0}))\hat{\tau}_{y}\Big{]},$
(22)

written in terms of local valley splitting $E_{\text{VS}}(x_{0})$ and the
local valley phase $\tilde{\varphi}_{\text{VS}}(x_{0})$.

The first is a linear gradient model (or smoothly tilted interface from [67])
depicted in Fig. 11d, in which the valley phase is given by
$\varphi_{\text{VS}}(x,y)=a_{x}x+a_{y}y$, where $a_{x}$ and $a_{y}$ denote
gradients along and perpendicular to the SQuS respectively. When substituted
to Eq. (VII.1), the parameters of effective gradient Hamiltonian
$\hat{H}_{\mathrm{v,g}}(x_{0})$ are given by:

$E_{\text{VS}}(x_{0})=E_{\text{VS},0}\exp[-\left(\frac{a_{x}L_{x}}{2}\right)^{2}],\,\,\tilde{\varphi}_{\text{VS}}(x_{0})=a_{x}x_{0},$
(23)

where $L_{x}\approx L_{\text{dot}}$ is the size of the QD in the x-direction.
The gradient in y-direction can be incorporated into definition of bare valley
splitting $E_{\text{VS},0}\equiv
E_{\text{VS},0}^{\prime}e^{-a_{y}^{2}L_{y}^{2}/4}$, where $L_{y}\approx
L_{\text{dot}}$ is the size of the QD in the y-direction.

In the second model we consider the modification of valley field
$\varphi_{\text{VS}}(x,y)$ caused by an atomistic step at the interface. We
consider regions $A_{n}$ having piecewise constant $\varphi_{\mathrm{VS},n}$,
see Fig. 11c. The Hamiltonian reads then:

$\hat{H}_{\mathrm{v,s}}(x_{0})=\frac{E_{\text{VS},0}}{2}\sum_{n}p_{n}(x_{0})\Big{[}\cos(\varphi_{\text{VS},n})\hat{\tau}_{x}+\sin(\varphi_{\text{VS},n})\hat{\tau}_{y}\Big{]}$
(24)

with the $p_{n}(x_{0})=\int_{A_{n}}\mathrm{d}A\,\rho(x-x_{0},y)$ being the
probability of the electron occupying region $A_{n}$. As a result, the valley
dynamics can be expressed in terms of the effective Hamiltonian (22), where
the $E_{\text{VS}}(x_{0})$ and $\tilde{\varphi}_{\text{VS}}(x_{0})$ are
indirectly defined via the equation:

$E_{\text{VS}}(x_{0})e^{i\tilde{\varphi}_{\text{VS}}(x_{0})}\equiv
E_{\text{VS,0}}\sum_{n}p_{n}(x_{0})e^{i\varphi_{\text{VS},n}},$ (25)

which mathematically represents the sum of the complex numbers, each
corresponding to $A_{n}$ region with respective modulus $p_{n}(x_{0})$ and
argument $\varphi_{\text{VS},n}$. We refer to this model as the step model.
When the electron travels over a single atomistic step the phase rotates by
$\varphi_{1}\equiv\varphi_{\text{VS},n+1}-\varphi_{\text{VS},n}\approx
0.85\pi$, and hence in the limit of separated steps, traveling electron will
experience local dips of valley splitting [67, 77, 104]. If the regions are
all separated by parallel steps, the y-confinement direction may be integrated
out and the model becomes effectively one dimensional with the missalignment
of the steps w.r.t. the shuttling direction $x$ entering as an effective
reduction of the shuttling velocity $\tilde{v}=v/\cos(\theta)$ with $\theta$
being the angle between the shuttling direction and the step normal, see Fig.
11c.

In order to parameterize the density of atomistic steps, the average tilt of
the Si/SiGe interface can serve as a reference parameter [105, 106],
originating from the miscut of the silicon wafer on which the heterstructures
is grown. A typical miscut angle of $\vartheta<1^{\circ}{}$ translates to an
average gradient of $a_{x}\approx 0.85\pi\vartheta/h$ or an average step
separation $\overline{d}\approx h/\vartheta$ with single atomic layer height
$h=a_{\mathrm{Si}}/4\sim 0.136$ nm. We note that the local gradients and step
densities may differ significantly from the global average (e.g. due to step
bunching or outliers in alloy disorder profile), and hence the miscut is only
taken as an indicator of order of magnitude of these effects.

###  VII.2 Excitation in the linear gradient model

To compute effects of non-zero gradient (Fig. 11d) from the model Hamiltonian
given in Eq. (22), we move to an adiabatic frame using a time-dependent
operator $\hat{R}(t)$ that diagonalizes the valley Hamiltonian
$\hat{R}(t)\hat{H}_{\mathrm{v,g}}(t)\hat{R}^{\dagger}(t)=\frac{1}{2}E_{\text{VS}}(t)\hat{\tau}_{x}$
at every instant of time [40]. When substituted into the Schrödinger equation,
it produces an effective Hamiltonian in the adiabatic basis
$\ket{e_{v}},\ket{g_{v}}$, which we assume to be the eigenstates of
$\hat{\tau}_{x}$ Pauli operator. Due to time-dependence of $\hat{R}(t)$ the
total effective Hamiltonian includes also the coupling between instantaneous
levels, which together reads:

$\hat{\mathcal{H}}_{\mathrm{v,g}}(t)=\hat{R}\hat{H}_{\mathrm{v,g}}(t)\hat{R}^{\dagger}-i\hbar\hat{R}\dot{\hat{R}}^{\dagger}=\frac{1}{2}E_{\text{VS}}\,\hat{\tau}_{x}+\frac{1}{2}\hbar\dot{\tilde{\varphi}}_{\text{VS}}\,\hat{\tau}_{z},$
(26)

where for each region of constant gradient,
$E_{\text{VS}}=E_{\text{VS},0}\exp(-[a_{x}L_{\text{dot}}]^{2}/4)$ and
$\dot{\tilde{\varphi}}_{\text{VS}}=a_{x}v$. As a result, the occupation of the
excited valley state is:

$\displaystyle
p_{e,v}(t)=|\bra{e_{v}}e^{-i\hat{\mathcal{H}}_{\mathrm{v,g}}t/\hbar}\ket{g_{v}}|^{2}$
$\displaystyle\quad=\frac{(\hbar a_{x}v)^{2}}{E_{\text{VS}}^{2}+(\hbar
a_{x}v)^{2}}\sin^{2}\left(\sqrt{E_{\text{VS}}^{2}+\hbar^{2}a_{x}^{2}v^{2}}\,\frac{t}{2\hbar}\right),$
(27)

which is the well known result of Rabi oscillations in the rotating frame. In
reality, instead of coherent oscillation, one should expect some time-
averaging due to inevitable fluctuation of electron velocity (see Fig. 7), or
valley-orbit coupling that allows for relaxation via phonon emission. Thus,
for every region of constant gradient we estimate typical occupation of
excited state as:

$\overline{p}_{e,v}\approx\frac{1}{2}\frac{(\hbar
a_{x}v)^{2}}{E_{\text{VS}}^{2}+(\hbar
a_{x}v)^{2}}\approx\frac{1}{2}\frac{(\hbar
a_{x}v)^{2}}{E_{\text{VS,0}}^{2}}\exp(\frac{(a_{x}L_{\text{dot}})^{2}}{2}),$
(28)

where the last approximation is valid if $\hbar a_{x}v\ll E_{\text{VS}}$,
which is fulfilled for typical gradients $a_{x}$ and parameters from Tab. 1.

###  VII.3 Excitation caused by sharp atomistic steps

Now, let us investigate the excitations caused by abrupt changes of
$\tilde{\varphi}_{\text{VS}}(x_{0})$ due to parallel atomistic steps at the
interface (Fig. 11c). For concreteness, we assume that the orientation of the
steps is perpendicular to the SQuS direction. The value of local valley
splitting reads
$E_{\text{VS}}(x_{0})=E_{\text{VS},0}\absolutevalue{\sum_{n}p_{n}(x_{0})e^{in\varphi_{1}}}$,
where $p_{n}$ is the probability of electron to be found in the $n$th
interstep region.

We start by computing the probability of the valley excitation on a single
atomistic step located at $x=0$. Assuming ground state electron density of the
form $\rho(x-vt)\propto e^{-(x-vt)^{2}/L_{\text{dot}}^{2}}$, we can write
probability of occupying regions on two sides of the step as
$p_{\text{L}(\text{R})}(t)=\frac{1}{2}(1-(+)\erf[vt/L])$, where erf($x$) is
the error function, using which we obtain

$E_{\text{VS}}(vt)=E_{\text{VS},0}\sqrt{\erf^{2}(vt/L)\sin^{2}\tfrac{\varphi_{1}}{2}+\cos^{2}\tfrac{\varphi_{1}}{2}}.$
(29)

In order to estimate the excitation probability on an isolated step, we use a
Landau-Zener model of non-adiabatic transition
$Q=\exp(-\pi\Delta^{2}/2\hbar\dot{\epsilon})$ [86], in which we identify time-
independent valley coupling as
$\Delta=E_{\text{VS},0}\cos\tfrac{\varphi_{1}}{2}$, and linearize time-
dependent part $\epsilon(t)\equiv
E_{\text{VS},0}\erf(\tfrac{vt}{L})\sin\tfrac{\varphi_{1}}{2}\approx\frac{2vtE_{\text{VS},0}}{L\sqrt{\pi}}\sin\tfrac{\varphi_{1}}{2}$.
The probability of occupying a higher valley state after single step passage
can then be written as

$Q_{1}=\exp{-\pi^{3/2}\frac{E_{\text{VS},0}\,L_{\text{dot}}}{4\hbar
v}\frac{\cos^{2}\tfrac{\varphi_{1}}{2}}{\sin\tfrac{\varphi_{1}}{2}}}=10^{-0.03/\eta}.$
(30)

In the last approximate expression, we have substituted the phase shift
corresponding to a single step, $\varphi_{1}=0.85\pi$, and combined the
remaining quantities into the shuttling parameter $\eta=\hbar
v/E_{\mathrm{VS},0}L_{\text{dot}}$. For a typical range of parameters
$v=10\,(50)$ m/s, $E_{\text{VS},0}=200\,\mu$eV, $L_{\text{dot}}=20$ nm, the
value of shuttling parameter $\eta\sim 0.002(0.01)$ corresponds to excitation
probabilities of the order of $Q_{1}\leq 10^{-17}\,(10^{-4})$. In Appendix D
we prove validity of L-Z approximation by a direct comparison against
numerical simulation.

Figure 12: Probability of occupying excited valley state, after two step
passage $Q_{2}$ for four different step separations $d$ corresponding to
double height step $d=0$ (green), valley collapse $d\approx
0.891L_{\text{dot}}$ (red) and interference $d\gg L_{\text{dot}}$ (blue) as a
function of shuttling parameter $\eta$ (lower scale) and electron velocity $v$
(upper scale) for fixed $E_{\text{VS},0}=200\mu$ eV and $L_{\text{dot}}=20$ nm
($E_{\text{orb}}=1$ meV). We added the intermediate case of $d=L_{\text{dot}}$
(black) to show that $d\geqslant L_{\text{dot}}$ distances between the steps
should be sufficient to avoid the valley collapse. In the interference regime
(blue) we compare analytical formulas for constructive interference
$Q_{2}^{\text{max}}=4Q_{1}$ (solid line) and the average $\langle
Q_{2}\rangle=2Q_{1}$ (dashed line) against the results of numerical simulation
(dots).

In the case of two parallel steps, located at $x=\pm d/2$, the $E_{\text{VS}}$
can identically vanish if the lateral distance between two steps fulfills
$E_{\text{VS}}(d)/E_{\text{VS},0}=\erf(d/2L_{\text{dot}})(1-\cos\varphi_{1})+\cos\varphi_{1}=0$,
which is numerically solved by $d\approx 0.891L_{\text{dot}}$. Using a
numerical simulation of evolution in Fig. 12 we show that the probability of
occupying the higher valley state after the two-step passage, $Q_{2}$, for the
step separation $d=0.891L_{\text{dot}}$ is close to unity even at small $v$.
This is caused by the above-mentioned a decrease of the energy gap due to the
competition between three regions of different valley phase. In contrast, this
is no longer true for slightly larger step separation $d=L_{\text{dot}}$
(black line), where the $Q_{2}$ starts to show signatures of interference
(note e.g. the dip around $\eta=10^{-2}$). The presence of interference is
even more visible at larger step separation i.e. $d>L_{\text{dot}}$, for which
the example result with $d=4L_{\text{dot}}$ is shown in Fig. 12 (blue line).
In this regime, the electron is expected to undergo consecutive transitions
through isolated atomistic steps, which coherently mix the two valley states.
Thus the whole process resembles Landau-Zener-Stückelberg interferometry [86].
In the case of two distant enough steps, the value of
$Q_{2}\sim|a_{1}+a_{2}|^{2}$ results from the coherent sum of probability
amplitudes, which correspond to excitation on the first
$a_{1}=\sqrt{Q_{1}}\sqrt{1-Q_{1}}$, and second step
$a_{2}=\sqrt{1-Q_{1}}\sqrt{Q_{1}}e^{i\Phi_{12}}$, where phase difference
between the two paths $\Phi_{12}$ gives:

$Q_{2}=4(1-Q_{1})Q_{1}\cos^{2}(\Phi_{12}/2)\approx
4Q_{1}\cos^{2}(\Phi_{12}/2).$ (31)

The last approximation works in the typical limit of $Q_{1}\ll 1$, see Eq.
(30). For estimation purposes one can consider the worst case scenario of
constructive interference ($\cos^{2}(\Phi_{12}/2)\to 1$), where
$Q_{2}^{\text{max}}\approx 4Q_{1}$. However, the typical occupation of the
excited valley state is expected to be closer to an average $\langle
Q_{2}\rangle\approx 2Q_{1}$, due to inevitable fluctuations of parameters that
randomize the phase $\Phi_{12}$ between the realizations
($\langle\cos^{2}(\Phi_{12}/2)\rangle\to 1/2$). In Fig. 12, we show an
agreement between analytical predictions (solid and dashed lines,
respectively) and numerical results (dots). The interference pattern is highly
sensitive to small variations of the electron velocity, and as such prone to
averaging in presence of expected velocity fluctuations (see Fig. 7).

Finally, let us generalize these results to the case of multiple interface
steps. Following [105, 106], the variation of the interstep distance is
expected to be proportional to the average distance between them,
$\sigma_{d}\sim\overline{d}$. Higher density of steps ($\overline{d}\ll
L_{\text{dot}}$) is expected to give more regular alignment, which in presence
of Ge interdiffusion at the interface [102] should resemble the linear
gradient model with relatively large $a_{x}=0.85\pi/\overline{d}$ (see the
previous Section). In the opposite limit of one or two atomistic steps per QD
size ($\overline{d}\geqslant L_{\text{dot}}$), their random alignment can lead
to both unfavourable valley-splitting collapse or favourable bunching of two
atomistic steps into one double-layer step. For the two-step case the bunching
can be modeled by Eq. (30) with $\varphi_{1}=1.7\pi$, and as shown using green
line in Fig. 12 it allows for adiabatic transfer even at relatively high
electron velocities.

We assume that the distance between all the steps is larger then the QD size,
$\overline{d}>L_{\text{dot}}$, which corresponds to the previously defined
interference regime. In analogy to the double step case, the probability of
occupying higher valley state after $N$-step passage, where
$N=L_{\mathrm{s}}/\overline{d}$ can be written as:

$p_{e,v}\approx
Q_{1}\left|\sum_{n=1}^{N}e^{i\Phi_{n}}\right|^{2}\approx\frac{L_{\mathrm{s}}}{\overline{d}}Q_{1}.$
(32)

where the approximation is valid for $NQ_{1}\leq 1$, and in the presence of
parameter fluctuations which result in uncorrelated sum of phases
$\sum_{n}e^{i\Phi_{n}}=\sqrt{N}$. The obtained result for multistep passage
will is tested against numerical solution in Appendix D.

###  VII.4 Valley relaxation due to electron-phonon coupling

The relaxation of the valley state in a gate defined QD may be treated
analogously to the spin relaxation in gate confined quantum dot systems [56,
53, 107], in that hybridization of valley and orbital leads to the dominant
relaxation mechanism [108, 109]. However, compared to the spin-orbit mediated
spin relaxation, the valley relaxation rate is higher due to both the stronger
valley-orbit coupling and the lack of Van-Vleck cancellation [110, 111, 112,
113].

As orbit-valley hybridization yields the dominant relaxation channel for
valleys, we may obtain the valley relaxation rates directly by substituting a
perturbative ansatz for the hybridized ground/excited valley states
$|\widetilde{g}_{v}\rangle$ and $|\widetilde{e}_{v}\rangle$, replacing
$|{g}_{o}\rangle$ and $|{e}_{o}\rangle$ in Eq. (16) and (17). This results in
the coupling matrix element
$|\langle\widetilde{g}_{v}|e^{i\bm{k}_{\lambda}\cdot\bm{\hat{r}}}|\widetilde{e}_{v}\rangle|^{2}\approx\left|\frac{\langle{e_{o}g_{v}}|H_{\mathrm{v}}|{g_{o}e_{v}}\rangle}{E_{\mathrm{orb}}}\right|^{2}|\langle
g_{o}|e^{i\bm{k}_{\lambda}\cdot\bm{\hat{r}}}|e_{o}\rangle|^{2}$ ($g_{o}$ and
$e_{o}$ and $g_{v}$ and $e_{v}$ labelling the ground/(first) exited orbital
and ground/exited valley states of the effective valley Hamiltonian
$H_{\mathrm{v}}$ from Eq. (22)). Neglecting some correction factor at maximum
of the order of unity and assuming $E_{\mathrm{VS}}\ll E_{\mathrm{orb}}$, we
obtain:

$\frac{1}{\tau_{\mathrm{r,v}}}=\left|\frac{\langle{e_{o}g_{v}}|H_{\mathrm{v}}|{g_{o}e_{v}}\rangle}{E_{\mathrm{orb}}}\right|^{2}F^{2}\left.\frac{1}{\tau_{\mathrm{r,o}}}\right|_{E_{\mathrm{orb}}\rightarrow
E_{\mathrm{VS}}},$ (33)

with
$F=1+\frac{E^{2}_{\mathrm{VS}}}{E^{2}_{\mathrm{orb}}}+\mathcal{O}\left(\frac{E^{4}_{\mathrm{VS}}}{E^{4}_{\mathrm{orb}}}\right)\approx
1$, and with $\tau_{\mathrm{r,o}}$ being the orbital relaxation time
calculated in Sec. VII. Neglecting correction factors, the coupling matrix
element is proportional to $E_{\mathrm{VS}}$, so that the relaxation rate
scales as $(E_{\mathrm{VS}}/E_{\mathrm{orb}})^{2}$. The orbital relaxation
rate evaluated at the valley splitting scales as
$\left.1/\tau_{\mathrm{r,o}}\right|_{E_{\mathrm{orb}}\rightarrow
E_{\mathrm{VS}}}\propto(E_{\mathrm{VS}})^{5}/E_{\mathrm{orb}}$ when neglecting
bottlenecking effects, which is approximately justified at the low valley
splitting energies of order $100\,\mathrm{\mu eV}$, for which in-plane
bottlenecking will just start to become relevant and out of plane
bottlenecking is sufficiently well suppressed
($k_{0}L_{z}=\frac{E_{\mathrm{VS}}}{\hbar v_{\lambda}}L_{z}<1$). With
$E_{\mathrm{orb}}$ roughly one order of magnitude higher than the valley
splitting, we would expect a $(E_{\mathrm{VS}}/E_{\mathrm{orb}})^{7}\approx
10^{-7}$ ratio of valley to orbital relaxation rates, if bottlenecking of the
orbital relaxation rates were negligible. However, as discussed in Sec. VI.4,
phonon bottlenecking has an significant effect on the orbital relaxation
rates, so that the ratio of valley to orbital relaxation rates is of the order
$10^{-5}-10^{-3}$ for relevant values. We note that the overall valley
relaxation rate scales as $E_{\mathrm{orb}}^{-3}$, so in principle sacrificing
QD confinement, while still ensuring QD stability could lead to an order of
magnitude increase in valley relaxation rate for realistic QD confinement
energies.

In order to estimate the orbit-valley hybridization we again employ the step
and gradient models. The gradient model (22) yields a matrix coupling element
of

$\displaystyle\langle
g_{v}|H_{\mathrm{v,g}}|e_{v}\rangle=\frac{a_{x}L_{\text{dot}}\,}{4\sqrt{\pi}}E_{\mathrm{VS,0}}e^{-(a_{x}L_{\text{dot}})^{2}/4}$

while the step model (24) yields

$\displaystyle\langle g_{v}|H_{\mathrm{v,s}}|e_{v}\rangle$
$\displaystyle=E_{\mathrm{VS,0}}\frac{\sin(\tfrac{\varphi_{1}}{2})e^{-\frac{\Delta
x^{2}}{L^{2}}}}{\sqrt{1+(p_{\mathrm{R}}-p_{\mathrm{L}})^{2}\tan^{2}\left(\frac{\varphi_{1}}{2}\right)}}$

where $\Delta x$ is the displacement from the QD center to the step position,
and $p_{L}$ ($p_{R}$) are the probabilities of the electron being to the left
(right) hand side of the step defined in Eq. (24).

Figure 13: Valley lifetimes for the stepmodel (a) and linear valley phase
model (b,c). The dashed lines in (a) are the inverses of the relaxation rate
averaged over the displayed region. A QD confinement energy of
$E_{\mathrm{orb}}=1\,\mathrm{meV}$ is assumed for all cases.

We use these two models to estimate $\tau_{\mathrm{r,v}}$, the results for
which are plotted in Fig. 13. For the step model, see Fig. 13a, the relaxation
rate is the largest when the QD center is between half to one QD length scale
($L_{\text{dot}}/2$ to $L_{\text{dot}}$) away from the position of the step.
While valley-orbit coupling is maximized if the QD is centered around the
step, the collapse in valley splitting significantly reduces the available
phonon modes at this frequency. The optimum for phonon assisted valley
relaxation is therefore slightly displaced. By moving further away from the
atomistic step, the overlap with the step exponentially decreases for a
Gaussian wavepacket, and gives rise to decrease in valley-orbit coupling, and
an effective suppression of this relaxation mechanism. A similar trade-off is
observed in the linear gradient model, see Fig. 13b: strong interface disorder
suppresses the valley splitting, while a lack of interface disorder prevents
valley-orbit hybridization, and therefore diminishes the relaxation. The
maximum relaxation rate occurs for gradients which change the valley phase by
$(0.1-0.2)\pi$ over one QD lengthscale $L_{\text{dot}}$. In terms of the
valley splitting, the maximum relaxation rate is achieved for valley
splittings close to the bare one ($E_{\mathrm{VS,0}}$), see Fig. 13c.

Overall, the achievable valley relaxation rate is highly dependent on the bare
valley splitting $E_{\mathrm{VS,0}}$. For our chosen ranges of parameters,
averaged valley relaxation times generally lie above $100\,\mathrm{ns}$,
which, as we have discussed in Sec. V, leads to randomization of the phase of
the spin qubit after an excitation event if the difference in spin-precession
frequencies between the different valley states leads to a significant buildup
of phase difference on this timescale. In conclusion, in order to avoid
detrimental effects due to the valley excitation, an increase in valley
splitting both suppresses non-adiabatic excitation as well as it increases the
valley relaxation rate due to phonons.

###  VII.5 Spin-valley hotspots

While shuttling the QD across a wide range of interface regions, the Zeeman
splitting may temporarily match the valley splitting $E_{Z}\approx
E_{\text{VS}}$. In such cases, spin relaxation is dominated by the spin-valley
hybridization, referred to as a spin-relaxation hotspot in the literature [52,
75, 56]. As such a relaxation channel leads to both spin-up decay ($\delta
p_{\uparrow}\equiv
p_{\uparrow}(0)-\Tr{\outerproduct{\uparrow}{\uparrow}\hat{\varrho}(\tau)}$) as
well as dephasing ($\delta C\sim\delta p_{\uparrow}$), but we will be simply
interested in the overall order of magnitude of the error $\delta
p_{\uparrow}$ in the following. Fortunately, this detrimental effect of spin-
valley hotspot can be completely avoided, if the Zeeman splitting is kept
below the global minimum or above the global maximum of the local valley
splitting. In particular at $B=20$ mT, the Zeeman splitting of $E_{z}\approx
2\,\mu$eV is well below the smallest value of local valley splitting reported
in the Si/SiGe quantum well, $E_{\text{VS}}\approx 10$ $\mu$eV [64]. Thus, the
use of small global magnetic fields is not only advantageous for suppressing
spin dephasing (Sec. IV), but also avoids spin relaxation at the spin-valley
hotspot.

However, if the presence of the hotspot cannot be prevented (e.g. due to local
collapse of valley splitting caused by unfavourable step separation or
relatively steep gradient), it will add to $\delta p_{\uparrow}$ and
consequently to $\delta C$. The hotspot is related to the avoided crossing
between the states $\ket{e_{v},\downarrow}$ and $\ket{g_{v},\uparrow}$, where
$g_{v}$ and $e_{v}$ label the ground and excited valley states, respectively.
Their energies are sketched in Fig. 14 together with the ground-state
$\ket{g_{v},\downarrow}$. There are two mechanisms contributing to $\delta
p_{\uparrow}$: (1) relaxation of the temporarily hybridized
$\ket{g_{v},\uparrow}$ and $\ket{e_{v},\downarrow}$ states to the ground state
$\ket{g_{v},\downarrow}$ which has the character of valley relaxation at the
hotspot, while effectively resulting in spin relaxation, (2) transfer of the
occupation from $\ket{g_{v},\uparrow}$ to $\ket{e_{v},\downarrow}$ state. The
latter process converts the spin qubit into a valley qubit, as it converts
superposition of spin states into that of valley states, provided that the
passage through the avoided crossing occurs via an adiabatic path. When this
happens, the quantum information carried by the qubit can be destroyed by
processes of relaxation to the valley ground state, and uncontrolled (due to
fluctuations of velocity and electric field noise) interference effects that
arise after the passing through the second anticrossing (see Fig. 14). For a
conservative estimate of the error we assume that taking an adiabatic path at
the first anticrossing results in complete decoherence of qubit state.

Figure 14: Sketch of energy difference between the following states: ground
valley spin-up state $\ket{g_{v},\uparrow}$, excited valley spin-down state
$\ket{e_{v},\downarrow}$ and the lowest-lying level $\ket{g_{v},\downarrow}$
with energy $E_{g}$ at the spin-valley hotspot caused by the atomistic step.
Inset: zoom into avoided crossing between $\ket{e_{v},\downarrow}$ and
$\ket{g_{v},\uparrow}$ states, where we identify two mechanisms responsible
for the spin-flip error $\delta p_{\uparrow}$: (1) the relaxation in vicinity
of avoided crossing, due to hybridisation between spin and valley states
(yellow arrow) and (2) adiabatic transition that converts spin up to spin
down, followed by subsequent relaxation, dephasing, and possibly more non-
adiabatic transitions at the subsequent level crossing (blue arrow). For
illustration purposes we have used arbitrary units.

For this conservative estimate, we concentrate on the adiabatic conversion
labeled as mechanism (2), since the contribution from (1) is about an order of
magnitude weaker, see Appendix. E for details. We use the Landau-Zener model
in the limit of fast passage, where the probability of going adiabatically
through the avoided crossing, that corresponds to
$\ket{g_{v},\uparrow}\to\ket{e_{v},\downarrow}$ transition, reads:

$\delta p_{\uparrow}^{(2,\text{art})}\leqslant
1-Q_{\text{LZ},\text{s-v}}\approx\frac{\pi}{2}\frac{\Delta_{\text{s-v}}^{2}}{\hbar
a}\to\frac{10^{-4}}{v[\mathrm{m/s}]},$ (34)

where we have used typical value of spin-valley mixing due to artificial-SOC
as $\Delta_{\text{s-v}}^{\text{art}}\sim 20$ neV, that have been estimated for
the gradient of $(\Delta B/\Delta x)=0.1\mathrm{mT/nm}$ (see Appendix. E), and
$a=d(E_{VS}(t)-E_{Z})/dt$. Larger gradients can significantly increase the
error since $\Delta_{\text{s-v}}^{\text{art}}\propto(\Delta B/\Delta x)$,
while for smaller ones the spin flip probability becomes dominated by
intrinsic-SOC mechanism of the similar order of magnitude $\delta
p_{\uparrow}^{(2,\text{int})}\\!\leqslant\\!10^{-4}/v\mathrm{[m/s]}$.
Naturally the gradient-dominated hotspots are most likely to occur in vicinity
of SQuS terminals, close to stationary qubits or in possible manipulation
regions, where most coherent control would take place. This shows that
avoiding strong gradients of magnetic field or reducing the value of constant
magnetic field to relatively hotspot-free $B=20$ mT is sufficient to avoid
shuttling error due to spin-valley hotspot.

###  VII.6 Transfer infidelity due to valley non-adiabatic effects

We conclude this section by combining the above calculation of valley
excitation and relaxation rates into an estimate for spin dephasing. We
initially consider relatively small bare valley splitting $E_{\text{VS},0}\leq
200$ $\mu$eV, which gives $\tau_{e,v}>\tau$ according to values from Fig. 13
and consequently falls into regime of no relaxation, in which $\delta C\approx
p_{e,v}$, see the discussion in Sec. V. In the smooth gradient model the final
occupation $p_{e,v}$ can be estimated by considering the largest gradient
along the SQuS. Assuming that $\hbar a_{x}v\ll E_{\text{VS},0}$, we have:

$p_{e,v}^{\text{grad}}\sim
10\left(\frac{a_{x}[\tfrac{1}{\text{nm}}]v[\tfrac{\text{m}}{\text{s}}]}{E_{\text{VS},0}[\mu\text{eV}]}\right)^{2}\exp(\frac{1}{2}(a_{x}\big{[}\tfrac{1}{\text{nm}}\big{]}L_{\text{dot}}[\text{nm}])^{2})\,\,.$
(35)

As the upper bound of the linear gradient, we take $a_{x}=0.05\pi/$ nm
$\approx\pi/L_{\text{dot}}$, which would correspond to a miscut angle of
$\vartheta=0.5^{\circ}{}$ and measurable local valley splitting
$E_{\text{VS}}=0.1E_{\text{VS,0}}\in[10,20]\,\mathrm{\mu eV}$. The most
pessimistic $E_{\text{VS,0}}=100$ $\mu$eV gives $p_{e,v}^{\text{grad}}\sim
10^{-4}\,(10^{-2})$ for $v=10\,(100)$ m/s, respectively, which shows that the
probability of valley excitations for the gradient values
$a_{x}\sim\pi/L_{\text{dot}}$ is below the threshold at $v\sim 10$ m/s. Note
that in presence of disordered interface, the considered value of
$a_{x}=\pi/L_{\text{dot}}$ can also emulate atomistic steps with average
separation $\overline{d}\approx 0.85L_{\text{dot}}$ (2-3 steps per QD size).

In contrast, the step model describes the valley dynamics for lower densities
of steps, $\overline{d}>L_{\text{dot}}$ (less then a step per QD size). The
local valley splitting $E_{\text{VS}}$ is then affected by a single step at a
time, and its value is not smaller then
$E_{\text{VS}}\geqslant\cos(0.85\pi)E_{\text{VS,0}}\sim
0.2E_{\text{VS,0}}\in[20,40]\mu$eV, which is in the lower range of commonly
measured local valley splittings. In this interference regime, we use Eq. (32)
to estimate:

$p_{e,v}^{\text{steps}}\approx\frac{10^{4}}{\overline{d}[\text{nm}]}\exp(-\frac{E_{\text{VS},0}[\mu\text{eV}]L_{\text{dot}}[\text{nm}]}{10\,v[\text{m}/\text{s}]}).$
(36)

For illustration, we take $\overline{d}=30$ nm (miscut angle $\vartheta\approx
0.25^{\circ}{}$), $E_{\text{VS},0}\\!=\\!100$ $\mu$ eV and $L_{\text{dot}}=20$
nm, which result in $p_{e,v}^{\text{steps}}\sim 10^{-7}\,(10^{-2})$ for
$v=10\,(20)$ m/s, respectively. The estimate reveals exponential sensitivity
of the excitation probability on a single step $Q_{1}$ to the shuttling
parameter $\hbar v/L_{\text{dot}}E_{\text{VS,0}}$, and in particular to the
electron velocity $v$.

Next, we consider the case of large bare valley splitting $E_{\text{VS}}$,
illustrated by the value of $E_{\text{VS},0}=500\,\mu\mathrm{eV}$, at which
the phonon relaxation becomes non-negligible. For the constant gradient model,
phonon relaxation is much slower than the rotation of the valley field and the
corresponding coherent oscillation. Hence, it mostly assists in reaching
average occupation of Eq. (VII.2). However, we highlight that larger values of
$E_{\text{VS},0}$ reduce the amplitude of oscillation and in this way limit
the phase error, i.e. $\delta C^{\text{grad}}\leqslant
p_{e,v}^{\text{grad}}\approx E_{\text{VS},0}^{-2}$. For the isolated step
model, Fig. 13 shows an estimate of the shortest valley relaxation time
$\tau_{e,v}\approx 100$ ns, which is only on the verge of reducing the phase
error $\delta C$ (see Sec. V). Indeed, for considered difference in valley-
dependent Larmor frequencies $\delta\omega_{v}/h\geqslant 0.5$ MHz, where the
equality holds for $B=20$ mT and $\delta g_{v}=10^{-3}$, one can estimate the
possible improvement in $\delta C$, i.e. the lower bound of the error, as:

$\delta C\approx
p_{e,v}\frac{\phi_{v}^{2}}{2}=\left(\frac{\delta\omega_{v}}{\hbar}\tau_{e,v}\right)^{2}p_{e,v}\geqslant
0.01p_{e,v}.$ (37)

This means that the presence of valley relaxation can realistically reduce
phase error by at most two orders of magnitude. Further improvement requires
shorter relaxation times (larger $E_{\text{VS},0}$ or stronger valley orbit
coupling), weaker magnetic fields, or smaller difference in valley-dependent
g-factors. We stress that in the step model, for $v\leq 40$ m/s and
$E_{\text{VS,0}}=500$ $\mu$eV, the valley relaxation is not really needed for
coherent operation of the SQuS, since corresponding probability of excitation
on a single step is negligibly small, $Q_{1}\sim 10^{-8}$, even for the
smallest considered QD size of $L_{\text{dot}}\sim 12$ nm ($E_{\text{orb}}=3$
meV).

##  VIII Discussion and Conclusion

In summary, we have provided a blueprint of a scalable Spin Qubit Shuttle
(SQuS) for quantum computers based on Si/SiGe quantum dots. The SQuS allows
for coherent communication over $\sim\\!10$ $\mu$m distance. We have argued
that this task can be achieved by moving a single electron spin qubit, using
one of two methods: consecutive charge transfers between pre-defined tunnel-
coupled QDs forming a chain - the Bucket Brigade (BB) mode - or keeping the
electron trapped in a moving quantum dot - the Conveyor Belt (CB) mode. The
scalability of both modes is provided by clavier gates with four-point
electric connection as proposed here, which avoid the signal fanout problems,
see Sec. II for details. In the same Section, we have argued that the BB mode