# Large spin shuttling oscillations enabling high-fidelity single qubit gates

Akshay Menon Pazhedath  Peter Grünberg Institute-Quantum Control (PGI-8),
Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany  Institute for
Theoretical Physics, University of Cologne, Zülpicher Straße 77, 50937
Cologne, Germany  Alessandro David  Peter Grünberg Institute-Quantum Control
(PGI-8), Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany  Max
Oberländer  JARA-FIT Institute for Quantum Information, Forschungszentrum
Jülich GmbH and RWTH Aachen University, Aachen, Germany  Matthias M. Müller
Peter Grünberg Institute-Quantum Control (PGI-8), Forschungszentrum Jülich
GmbH, D-52425 Jülich, Germany  Tommaso Calarco  Peter Grünberg Institute-
Quantum Control (PGI-8), Forschungszentrum Jülich GmbH, D-52425 Jülich,
Germany  Institute for Theoretical Physics, University of Cologne, Zülpicher
Straße 77, 50937 Cologne, Germany  Dipartimento di Fisica e Astronomia,
Università di Bologna, 40127 Bologna, Italy  Hendrik Bluhm  JARA-FIT Institute
for Quantum Information, Forschungszentrum Jülich GmbH and RWTH Aachen
University, Aachen, Germany  ARQUE Systems GmbH, 52074 Aachen, Germany  Felix
Motzoi  Peter Grünberg Institute-Quantum Control (PGI-8), Forschungszentrum
Jülich GmbH, D-52425 Jülich, Germany

###### Abstract

Semiconductor quantum dots have shown impressive breakthroughs in the last
years, with single and two qubit gate fidelities matching other leading
platforms and scalability still remaining a relative strength. However, due to
qubit wiring considerations, mobile electron architectures have been proposed
to facilitate upward scaling. In this work, we examine and demonstrate the
possibility of significantly outperforming static EDSR-type single-qubit
pulsing by taking advantage of the larger spatial mobility to achieve larger
Rabi frequencies and reduce the effect of charge noise. Our theoretical
results indicate that fidelities are ultimately bottlenecked by spin-valley
physics, which can be suppressed through the use of quantum optimal control,
and we demonstrate that, across different potential regimes and competing
physical models, shuttling based single-qubit gates retain significant
advantage over existing alternatives.

##  I Introduction

Spin-based devices in semiconducting heterostructures Burkard  _et al._ (2023)
are promising candidates to host and manipulate qubits Loss and DiVincenzo
(1998) as they offer advantages like inherent two-level structure,
controllability through gate-tunable electrostatic potentials and direct
interface with the semiconductor industry Klemt  _et al._ (2023); Zwerver  _et
al._ (2022). Si/SiGe heterostructures have been widely used to harbor spin
qubits in quantum dots, with long spin relaxation time (beyond seconds)Simmons
_et al._ (2011) and dephasing time (beyond tens of microseconds)Kawakami  _et
al._ (2014); Neumann and Schreiber (2015). Moreover, semiconductor quantum
dots have shown impressive breakthroughs in the last years, with single-qubit
and two-qubit gate fidelities matching other leading platforms Yoneda  _et
al._ (2018); Xue  _et al._ (2022); Mills  _et al._ (2022); Noiri  _et al._
(2022).

Scalability of such systems is limited by the short range of the exchange
interaction, which leads to a fan-out of electrostatic gates when wiring a
large number of qubits, as well as to crosstalk errors. Sparse architectures
with on-chip controls Vandersypen  _et al._ (2017); Boter  _et al._ (2022);
Künne  _et al._ (2023) are among the possible solutions, which require a long-
range coherent link between distant qubit registers. We focus here on systems
where the qubit is moved by a series of electrostatic gates allowing bi-
directional conveyor-mode shuttling Seidler  _et al._ (2021); Langrock  _et
al._ (2023); Xue  _et al._ (2023); Struck  _et al._ (2023). In this approach,
the spin is manipulated by transporting it to a dedicated zone where a micro-
magnet is present.

Single-qubit operations of quantum dot spins in presence of an artificial
spin-orbit field have already been demonstrated Nowack  _et al._ (2007);
Pioro-Ladrière  _et al._ (2008); Yoneda  _et al._ (2014); Kawakami  _et al._
(2014); Yoneda  _et al._ (2018). These experiments leverage electric-dipole
spin resonance (EDSR) Golovach  _et al._ (2006), where the quantum dot
confinement potential is oscillated beneath a static micro-magnet in order for
the spin to experience a periodic magnetic field. However, this is achieved in
a setting where the amplitude of oscillation is limited to
$\sim$$1\text{\,}\mathrm{nm}$. This leads to larger gradient field strength
requirements, which cause spin dephasing.

In contrast, shuttling architectures such as the SpinBus architecture Künne
_et al._ (2023) is predicted to have the capability of performing larger
amplitude oscillations, up to $10$ or $20\text{\,}\mathrm{nm}$, weakening the
gradient field requirement by at least an order of magnitude, and improving
the spin dephasing time. This can be combined with a geometry-optimized micro-
magnet for further improvements Yoneda  _et al._ (2015); Dumoulin Stuyck  _et
al._ (2021).

An obstacle to achieve high-fidelity gates in this context is represented by
the presence of low lying valley states in the conduction band minimum of Si
Zwanenburg  _et al._ (2013). Evidence shows that these states couple to the
spin degree of freedom (DOF) by having different g-factors Ruskov  _et al._
(2018); Ferdous  _et al._ (2018); Veldhorst  _et al._ (2015); Rančić and
Burkard (2016). This modifies the standard fixed-frequency Rabi formula by
introducing spatially dependent energy shifts, a fact that complicates the
controllability and may ultimately lead to decoherence. The interaction
between the electron wavefunction and the heterostructure walls lifts the
degeneracy of the valley DOF, but the splitting is very sensitive to the local
atomic arrangement Wuetz  _et al._ (2022); Friesen  _et al._ (2007). As the
quantum dot is shuttled, the valley DOF may spend an undetected amount of time
in its excited state, corresponding to a shift in the precession rate of the
spin and to phase randomization.

In this work we model the shuttling-based EDSR operation, taking into account
the dephasing mechanism between spin and valley states. A conveyor-mode
shuttling device is modelled by generating position dependent valley
eigenstates and eigenvalues, termed as valley landscape throughout this work,
according to two distinct models. One model considers the presence of Ge atoms
in the Si well (termed as the ‘Ge-diffusion model’), while the other focuses
on the presence of fabrication miscuts along the length of the device (termed
as the ‘step model’)A. David et al. (2024). We show that the large electron
oscillations increase the spin dephasing time of the qubit while operating at
an order of magnitude lower magnetic gradient, with single-qubit gate fidelity
limited by low valley splitting points
$<$$15\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ (LVSPs) and large spin-valley
coupling. The electron trajectory can be shaped using optimal control
techniques presented in this work to improve the fidelity by at least two
orders of magnitude, in a spin-valley physics agnostic manner. We study the
dependence of the average gate fidelity on the driving frequency and the
spatial amplitude of the trajectory to understand underlying physical effects,
and we optimize over a parameter grid of pulse lengths and spatial amplitudes
to show the advantage of optimal control in realizing fast high-fidelity
gates. An extensive comparison of the valley models discussed above for
various realistic scenarios, in terms of attainable fidelities, is also
performed to elaborate on the advantage of using trajectory shaping over other
methods. Our results indicate that shuttling-based EDSR is a viable and
preferable way of performing single qubit operations in semiconducting
platforms.

This paper is structured as follows. In Sec. II, we explain the relevant
details of the SpinBus architecture and the model Hamiltonian used for our
simulations, with a description of the different valley models considered. We
follow this with a discussion on implementing the dynamics and optimization of
the single-qubit operation, where we elucidate the optimal control methodology
used. Sec. III deals with the main results of this work, where we show the
advantages of a large-amplitude driven EDSR and compare the valley models
based on the gate fidelities for a single prototypical device as well as for
$1000$ different sampled devices to ascertain the advantage of using optimal
control in shaping electron trajectories. We discuss potential obstacles and
future works in Sec. IV and conclude in Sec. V.

Figure 1: Schematic of the manipulation zone in SpinBus. (a) A suitable micro-
magnet is placed on top of the electrostatic gates (same color gates receive
the same phase shifted sinusoidal signal), which enables shuttling based
single-qubit gates for the electron spin (blue ball) trapped in the Si quantum
well. The trapping potential (shown in red) is moved back and forth to induce
EDSR for the spin. The interface of SiGe and Si can be modelled in two
different ways, as shown in (b) and (c).

##  II Conveyor-mode single qubit rotations

In this section, we model the manipulation zone of the quantum bus
architecture (see Fig. 1) and we discuss the simulation and optimization of
the single-qubit operation.

###  II.1 Device architecture

In a conveyor-mode spin-shuttling device Langrock  _et al._ (2023), a row of
electric gates generates an array of quantum dots with sinusoidal profile
inside the quantum well of a Si/SiGe heterostructure. The transported qubit is
encoded in the spin of a single electron trapped in one of these quantum dots.
Bi-directional shuttling of the electron over
$\sim$$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is possible Xue  _et al._
(2023) with only four phase shifted sine waves as control signals to the
gates. Fig. 1 illustrates such an apparatus.

Single qubit gates are performed via EDSR at the so called manipulation zone ,
by oscillating the electron under an external magnetic field ($\sim$[range-
units = single, range-phrase=–]2050) Künne  _et al._ (2023) around a magnetic
gradient generated by a suitable micro-magnet. In comparison, the high
fidelity gates shown by Ref. Yoneda  _et al._ (2018) adopt a similar approach
of using micro-magnets in conjunction with electron position oscillation to
perform EDSR. The difference is in the choice of the operating regime, with a
far higher external magnetic field ($\sim$$0.5\text{\,}\mathrm{T}$), an order
of magnitude higher gradient magnetic fields and an order of magnitude lower
amplitude for oscillations.

###  II.2 Model Hamiltonian

We build on the model presented in Ref. A. David et al. (2024) by adding an
extra term describing the artificial spin-orbit field. The computational
subspace of interest is the spin of the electron in the quantum dot, a two-
level system (TLS). The spin can interact with the so called valley degree of
freedom (DOF) in Si, also a TLS Zwanenburg  _et al._ (2013). The combined
Hilbert space can be written as

$\mathcal{H}=\mathcal{H}_{\text{valley}}\otimes\mathcal{H}_{\text{spin}},$ (1)

with $\ket{+k_{z}},\ket{-k_{z}}$ as the valley basis and ${\ket{0},\ket{1}}$
(computational basis) as the spin basis. Here, $\ket{+k_{z}},\ket{-k_{z}}$
correspond to the two low-lying valley states after the $6$-fold degeneracy in
the conduction band of Si is lifted by strain.

The Hamiltonian has four terms,

$H=H_{\text{Zeeman}}+H_{\text{Rabi}}+H_{\text{valley}}+H_{\text{spin-
valley}},$ (2)

where

$H_{\text{Zeeman}}=\frac{1}{2}g\mu_{B}B_{z}~{}\mathds{1}\otimes\sigma_{z}$ (3)

with $g$ being the Landé g-factor of the electron ($\approx$$2$), $\mu_{B}$
being the Bohr magneton, $B_{z}$ being the constant external magnetic field
applied across the manipulation zone and $\sigma_{z}$ is the $z$ Pauli matrix
acting on the spin. The value of $B_{z}$ is chosen to be
$20\text{\,}\mathrm{mT}$ in this work. The drive term,

$H_{\text{Rabi}}=\frac{1}{2}g\mu_{B}B_{x}(x_{\text{qd}}(t))~{}\mathds{1}\otimes\sigma_{x}$
(4)

corresponds to the perpendicular magnetic field
$B_{x}(x_{\text{qd}}(t))=\partial b_{\perp}\cdot x_{\text{qd}}(t)$ due to
micro-magnets, responsible for Rabi driving. Here, $\sigma_{x}$ is the $x$
Pauli matrix acting on spin and $x_{\text{qd}}(t)$ is the position of the QD
center at time $t$. The value of $\partial b_{\perp}$ for the SpinBus is
typically chosen to be $0.1\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$. We
consider at first the position of the electron varied as a simple sine wave
given as

$x_{\text{qd}}(t)=x_{0}\sin{(\omega t+\phi)}.$ (5)

with $x_{0}$, $\omega$ and $\phi$ denoting the amplitude, frequency and phase
of the oscillation respectively. The SpinBus architecture provides dynamic
control over the position $x_{\text{qd}}$ of the trapping potential, using the
same finger gates responsible for shuttling the electron through the device,
which directly translates into spatial control of the electron trapped inside
the potential well. $H_{Zeeman}$ causes precession of the spin around the
quantization axis, and $H_{Rabi}$ acts perpendicular to the quantization axis
to rotate the spin in the preferred direction. The choice of the phase in Eq.
5 gives the usual $\sigma_{x}$ and $\sigma_{y}$ gates in the rotating frame.
Rabi oscillations occur between the spin up and spin down states due to EDSR,
with the frequency

$\omega_{\text{Rabi}}=g\mu_{B}\partial b_{\perp}x_{0}/\hbar,$ (6)

which is modified as $\Omega=\sqrt{\delta^{2}+\omega_{Rabi}^{2}}$ when there
is a detuning $\delta$. In general, the valley Hamiltonian can be expressed
as:

$H_{\text{valley}}=\Delta_{\text{real}}(x_{\text{qd}}(t))\tau_{x}+\Delta_{\text{imag}}(x_{\text{qd}}(t))\tau_{y}$
(7)

where $\tau_{x}$ and $\tau_{y}$ are Pauli operators in the valley subspace,
$\Delta_{\text{real}}(x_{\text{qd}}(t))$ and
$\Delta_{\text{imag}}(x_{\text{qd}}(t))$ corresponds to the real and imaginary
part of the complex position-dependent intervalley coupling. The magnitude of
valley splitting is given by
$E_{V}(x_{\text{qd}})=2\sqrt{\Delta_{\text{real}}^{2}(x_{\text{qd}}(t))+\Delta_{\text{imag}}^{2}(x_{\text{qd}}(t))}$,
and the valley phase by
$\varphi_{V}=\arg(\Delta_{\text{real}}(x_{\text{qd}}(t))+i\Delta_{\text{imag}}(x_{\text{qd}}(t)))$.
The values of $\Delta_{\text{real}}(x_{\text{qd}}(t))$ and
$\Delta_{\text{imag}}(x_{\text{qd}}(t))$ are generated according to two
different models for the microscopic valley physics (cf. Fig. 1 (b) and (c),
respectively). The Ge-diffusion model considers the microscopic arrangement of
Ge atoms inside the Si well, leading to larger or smaller valley splittings
depending on the overlap of the electron wavefunction with Ge atoms. On the
other hand, the step model describes the effect of interface miscuts arising
from the growth process of the heterostructure. This model assumes the region
between two miscuts to be smooth and to have a uniform intervalley coupling,
but the presence of a miscut changes the effective ground and excited state of
the valley before and after the miscut. An extensive treatment of both models
can be found in A. David et al. (2024).

The last term in the Hamiltonian is a position dependent effective ZZ-
interaction between the valley and spin DOF, given by

$H_{\text{spin-
valley}}=-\kappa_{z}\left(\tilde{\tau}_{z}(x_{\text{qd}})\otimes\sigma_{z}\right)$
(8)

where $\kappa_{z}$ is typically chosen to be on the order of [range-
units=single,range-phrase=–]1e-65e-6, which is proportional to the relative
$g$-factor variation $\delta g/g$ Ruskov  _et al._ (2018); Rančić and Burkard
(2016). Here,
$\tilde{\tau}_{z}(x_{\text{qd}})=\\{\cos(\varphi_{V}(x_{\text{qd}}))\tau_{x}+\sin(\varphi_{V}(x_{\text{qd}}))\tau_{y}\\}$
is the $z$ Pauli matrix acting on the ground and excited states of valley, and
$\varphi_{V}(x_{\text{qd}})$ is the valley phase at a chosen starting
position.

###  II.3 Dynamics and Optimization

The combined spin-valley system follows open system dynamics, with the valley
DOF decaying in contact with the crystal environment and the spin DOF
dephasing because of charge noise. The density matrix of the combined system,
$\rho(t)$, evolves according to the master equation

$\frac{\mathrm{d}\rho}{\mathrm{d}t}(t)=-\frac{i}{\hbar}[H\left(x_{\text{qd}}(t)\right),\rho(t)]\\\
+\frac{1}{T_{1,v}}\mathcal{D}\left[\tilde{\tau}_{-}\left(x_{\text{qd}}(t)\right)\right](\rho(t))+\frac{1}{T_{2,s}}\mathcal{D}[\sigma_{z}](\rho(t)),$
(9)

where we have the dissipative operator $\mathcal{D}[L](\rho)=L\rho
L^{\dagger}-(\rho LL^{\dagger}+LL^{\dagger}\rho)/2$,
$T_{1,v}=~{}$$100\text{\,}\mathrm{ns}$ (corresponding to valley decay), and
$T_{2,s}=$ [range-units=single, range-phrase=–]2080 (corresponding to spin
dephasing). The jump operator used for the valley relaxation is given by A.
David et al. (2024)

$\tilde{\tau}_{-}(x_{\text{qd}})=\begin{pmatrix}1&e^{-i\varphi_{V}(x_{\text{qd}})}\\\
e^{i\varphi_{V}(x_{\text{qd}})}&-1\end{pmatrix},$ (10)

which is the lowering operator from the local valley excited state to the
local valley ground state.

####  II.3.1 GRAPE with JAX

The trajectory of electron as shown in Eq. 5 is discretized into piecewise-
constant functions as in the usual approach of the well known GRAPE algorithm
Khaneja  _et al._ (2005). The time step is chosen so as to capture the
relevant spin-valley dynamics, and is calculated to be around
$0.8\text{\,}\mathrm{ps}$ from the average valley splitting along the length
of the device. The discretized trajectory is used to perform state evolution
and finally the average gate fidelity is calculated as explained in Sec.
II.3.3. Performance improvements for state evolution and fidelity evaluation
are achieved using the JAX library in python James Bradbury and Roy Frostig
and Peter Hawkins and Matthew James Johnson and Chris Leary and Dougal
Maclaurin and George Necula and Adam Paszke and Jake VanderPlas and Skye
Wanderman-Milne and Qiao Zhang (2018). This also enables auto-differentiation
of the cost function, which is essential for back-propagating the errors to
perform optimization. The gradients found using auto-differentiation are used
with the L-BFGS optimization method in SCIPY Virtanen  _et al._ (2020), to
minimize the average gate infidelity ($1-\bar{F}$). The optimization is done
for a lesser number of controls (fixed to $10$ controls per ) than the time-
discretized steps, and then interpolated while calculating the final gate
fidelity Motzoi  _et al._ (2011).

####  II.3.2 Shaping function

The initial control pulse is sinusoidal, as in Eq. 5. To ensure that the pulse
starts and ends at the same point, we shape the pulse using a Gaussian ramp
envelope given as

$G(t)=\begin{cases}f_{s}(t-t_{r}),&\text{if}~{}t\leq t_{r}\\\
1,&\text{if}~{}t\geq t_{r}\wedge t\leq T_{g}-t_{r}\\\
f_{s}(\tilde{t}+t_{r}),&\text{if}~{}t\leq T_{g}-t_{r}\end{cases}$ (11)

where
$f_{s}(t)=\alpha[\exp(-t^{2}/2\sigma^{2})-\exp(-t_{r}^{2}/2\sigma^{2})]$, with
$\tilde{t}=t-t_{f}$, $\sigma=t_{r}/4$ and
$\alpha=[1-\exp(-t_{r}^{2}/2\sigma^{2})]^{-1}$. Here $t_{r}$ is the rise time,
set here to $1\text{\,}\mathrm{ns}$, and $T_{g}$ is the gate time. The shaped
pulse to be optimized will be $\tilde{x}_{\text{qd}}(t)=G(t)\cdot
x_{\text{qd}}(t)$.

####  II.3.3 Cost function

The function to be maximized is the average gate fidelity of the quantum
channel $\mathcal{E}$, evolving an initial spin state according to Eq. 9 and
subsequently tracing out the valley DOF, compared to a desired unitary $U$. We
use

$\bar{F}(\mathcal{E},U)=\frac{dF_{\text{ent}}(\mathcal{E},U)+1}{d+1},$ (12)

as defined in Nielsen (2002), with $d=2$. Here,
$F_{\text{ent}}(\mathcal{E},U)$ is the entanglement fidelity and is denoted by
$F_{\text{ent}}(\mathcal{E},U)=\bra{\phi}(\mathds{1}\otimes\,\mathcal{U}^{\dagger}\circ\mathcal{E})(\phi)\ket{\phi}$,
where $\ket{\phi}$ is a maximally entangled state of the spin with an ancilla
qubit and $\mathcal{U}^{\dagger}(\rho)=U^{\dagger}\rho U$. In practice, the
calculation of $F_{\text{ent}}(\mathcal{E})$ involves the propagation of four
initial spin states that are orthogonal in the Bloch sphere.

For our simulations, the target gate $U_{G}$ was chosen to be a rotation of
$\pi$ about the y-axis, given by

$U_{G}=\exp\left(-i\pi\sigma_{y}/2\right).$ (13)

As the simulated evolution in Eq. 9 is in the lab frame, the spin precesses
under the action of $H_{\text{Zeeman}}$ and $H_{\text{spin-valley}}$, in
addition to the desired gate. In the ideal case of the electron remaining
always in the valley ground state throughout the evolution, the frequency of
precession is $\hbar\Omega_{R}=2\mu_{B}B_{z}+2\kappa_{z}$. We counter-rotate
the final spin state to transform into a precession-free rotating frame via
$U_{R}=\exp{\left(-i\Omega_{R}\sigma_{z}\right)}$. Finally in Eq. 12 we use
$U=U_{R}U_{G}$ which ensures that the optimizer will favor the trajectory that
keeps the electron in the valley ground state.

Figure 2: Effect of transverse magnetic gradient on spin dephasing and average
gate fidelity for varying valley splittings. Ge-diffusion model is used to
generate a prototypical device, and three points along the device are sampled.
Pulses with varying $x_{0}$ (and corresponding analytical $T_{g}$) are used to
obtain the average gate infidelity ($1-\bar{F}$) at each point. The minimum of
$1-\bar{F}$ with respect to $x_{0}$ (indicated by *), for varying $\partial
b_{\perp}$ and corresponding spin dephasing rates are plotted here. (a-b)
Large valley splitting helps at retaining good fidelity as spin dephasing rate
increases (red line). (c) LVSP combined with large spin-valley coupling leads
to an order of magnitude worse average gate fidelity (red line with
triangles). In all cases, the advantage of shuttling the qubits over operating
them static can be seen by comparing their infidelities, as indicated by the
green ($\partial b_{\perp}=$ $0.1\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$) and
red ($\partial b_{\perp}=$ $1\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$) stars
respectively. Irrespective of the competing physical mechanisms causing
decoherence, optimal control always converges below ${10}^{-3}$, with
significant advantage from shuttling (teal dash-dot line).

##  III Results

This section is divided into three parts. We first compare shuttling-based
EDSR with static EDSR, by relating the influence of transverse magnetic field
gradients on spin-dephasing and average gate fidelity. Next, we investigate
the spin-valley interplay mechanism and the use of optimal control as a mean
to perform high fidelity gates for driving pulses ranging in amplitude and
duration. Finally, we show the differences in single-qubit gate fidelity for
competing valley models. Device-specific and ensemble-averaged gate fidelities
are analysed to understand the limitations imposed by spin-valley decoherence.

Figure 3: Trajectory parameters and optimization. Amplitude of the driving
pulse is swept along with (a) driving frequency and (b) pulse length to
calculate the average gate infidelity, for a device simulated using Ge-
diffusion model with spin-valley coupling $\kappa_{z}=$
$5\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$. (a) left panel shows the case
with no spin-valley coupling. When the driving frequency is closer to the Rabi
frequency, higher fidelity is achieved. In the presence of spin-valley
coupling the Rabi frequency is modified (visible from the upwards shift), and
LZSM interference of valley excited states leads to fringes, as shown in the
right figure of (a). The distinctive Chevron patterns are visible in (b), with
no parameter combinations yielding infidelities below ${10}^{-3}$ (left
panel). Pulses obtained through optimal control performs better at preserving
the average gate fidelity, minimizing infidelities to well below ${10}^{-3}$
(right panel).

###  III.1 Static vs. shuttled quantum dots: Effect on spin dephasing

Static quantum dots in a Si/SiGe heterostructure have achieved high fidelity
single-qubit operations using micro-magnet enabled EDSR, with a $T_{2}^{*}=$
$20\text{\,}\mathrm{\SIUnitSymbolMicro s}$ Yoneda  _et al._ (2018), while
operating in the regime mentioned in Sec. II.1. EDSR based on lower amplitudes
limits achievable gate time, resulting in slower gates. In principle, larger
$x_{0}$ combined with large $\partial b_{\perp}$ could lead to faster gates,
but keeping small gradients reduces the effects of charge noise.

Charge noise arising from voltage fluctuations in confinement potentials is
transferred to the spin as dephasing noise by the longitudinal component of
the magnetic gradient, $\partial b_{\parallel}$. This reduces the $T_{2}^{*}$
of a device, which is derived in Ref. Kawakami  _et al._ (2014) as
$T_{2}^{*}=2\sqrt{\ln(2)}/(\pi\delta f_{\text{FWHM}})$, where $\delta
f_{\text{FWHM}}=g\mu_{B}\partial b_{\parallel}\delta x_{\text{rms}}/h$, with
$\delta x_{\text{rms}}$ denoting the root mean square displacement of the
electron position $x$ caused by voltage fluctuations (see Ref. Yoneda  _et
al._ (2018) supplementary). This yields

$T_{2}^{*}=\frac{2h\sqrt{\ln(2)}}{\pi g\mu_{B}\partial b_{\parallel}\delta
x_{\text{rms}}}.$ (14)

Changing the geometry of the micro-magnet can maximize the ratio $Q=\partial
b_{\perp}/\partial b_{\parallel}$, between transverse and longitudinal
gradients, to improve $T_{2}^{*}$ of a device Dumoulin Stuyck  _et al._
(2021). Although design-dependent, maximum and minimum values of $Q$ are in
general finite and in the range of one order of magnitude. It follows that the
choice of transverse gradient used for EDSR will influence spin dephasing.
Substituting $\partial b_{\parallel}=\partial b_{\perp}/Q$ in Eq. 14 we obtain

$T_{2}^{*}=\frac{2hQ\sqrt{\ln(2)}}{\pi g\mu_{B}\partial b_{\perp}\delta
x_{\text{rms}}}=\frac{4QT_{g}x_{0}\sqrt{\ln(2)}}{\pi\delta x_{\text{rms}}},$
(15)

directly relating the $T_{2}^{*}$ with $Q$. The amplitude $x_{0}$ and the
analytical gate time $T_{g}$ computed as

$T_{g}=\frac{\pi}{\omega_{Rabi}}=\frac{\pi\hbar}{g\mu_{B}\partial
b_{\perp}x_{0}}\,.$ (16)

In Ref. Yoneda  _et al._ (2018), it is estimated that $\delta x_{\text{rms}}=$
$4\text{\,}\mathrm{pm}$, and the quantum dot is operated at $\partial
b_{\perp}=$ $1\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$ and $\partial
b_{\parallel}=$ $0.2\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$. In order to
study the effect of $\partial b_{\perp}$ on $T_{2}^{*}$, we relate these
quantities by fixing $Q=5$. In the case of EDSR by shuttling, as one reaches
an order of magnitude higher amplitudes, the transverse gradient can be
$\partial b_{\perp}=$ $0.1\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$ Langrock
_et al._ (2023) and therefore we assume $\partial b_{\parallel}=$
$0.02\text{\,}\mathrm{mT}\text{/}\mathrm{nm}$.

We use the Ge-diffusion model to generate a valley landscape, and base our
study on two different spin-valley couplings, specifically $\kappa_{z}=$
$1\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$(typically expected value Ruskov
_et al._ (2018)) and $\kappa_{z}=$
$5\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$. Three points are sampled from
this valley landscape, corresponding to a valley splitting of
$54.53\text{\,}\mathrm{\SIUnitSymbolMicro eV}$,
$23.52\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ and
$2.61\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ respectively. For each point,
$\partial b_{\parallel}$ is varied from [per-mode=symbol,range-units=single,
range-phrase=–]0.020.2, yielding the corresponding $\partial b_{\perp}$ and
$T_{2}^{*}$ (see Eq. 14). It follows from Eq. 6 that the Rabi frequency is
directly proportional to $\partial b_{\perp}x_{0}$. For each value of
$\partial b_{\perp}$, we vary the analytical $T_{g}$ from [range-units=single,
range-phrase=–]5360 to generate electron trajectories with the corresponding
analytical $x_{0}$ and calculate the average gate infidelity. From this, the
minimum of the infidelity, $(1-\bar{F})^{*}$, with respect to $x_{0}$ is
calculated (Fig. 2, brown solid line). The same procedure is repeated for
optimized electron trajectories (as explained in Sec. II.3) corresponding to
each $T_{g}$ and $x_{0}$ (Fig. 2, teal dash-dot line).

The following observations are in order: First, smaller magnetic gradients
results in smaller spin dephasing rates for relatively larger values of valley
splittings as shown by the brown solid lines in Fig. 2(a)-(b). The advantage
of smaller gradients wanes at a significantly lower valley splitting, denoting
the importance of spin-valley physics at larger amplitudes, as shown in Fig.
2(c). The effect of larger spin-valley couplings only shows a noticeable
difference in the case of a small valley splitting. Second, optimal control of
electron trajectories ensures an average gate infidelity below ${10}^{-3}$,
irrespective of the magnitude of the valley splitting or spin-valley coupling
for all magnetic gradients.

Figure 4: Dependence of average gate fidelity on micro-magnet position in the
manipulation zone. For a device simulated using the Ge-diffusion model, the
position of micro-magnet is varied over a chosen length. (a) Valley splitting
dependence on position. (b) Analytical (brown line),
($\omega,T_{g}$)-optimized (orange dashed line) and trajectory optimized (teal
dash-dot line) infidelity as a function of position, for $\kappa_{z}=10^{-6}$
meV. As the valley splitting encountered decreases, the infidelity increases
and becomes worse when it encounters a LVSP. Optimal control improves the
average gate fidelity even at hard-to-navigate LVSPs. (c) For
$\kappa_{z}=5\times 10^{-6}$ meV, this effect is enhanced for the analytical
and ($\omega,T_{g}$)-optimized lines, while optimal control succeeds at
retaining high fidelity. (d), (e) and (f) shows the same for the step model of
valley splitting.

Figure 5: Optimal control improves average gate fidelity over $1000$ sampled
devices. Ge-diffusion model and step model are used to generate 1000 different
shuttling devices of $200\text{\,}\mathrm{nm}$ length, with varying valley
splitting distributions. (a) Valley splitting distribution calculated at the
center of each device (at $100\text{\,}\mathrm{nm}$). (b) Bar plots between
average gate infidelity and valley splitting with mean marked for Ge-diffusion
model (top row) and step model (bottom row). Increasing values of spin-valley
coupling leads to poor average gate fidelity for LVSPs, as shown by the
analytical infidelity bars (in brown). ($\omega,T_{g}$) optimization can be
employed to improve fidelity for larger valley splittings, but is limited by
LVSPs (orange bars). Pulse engineering helps in improving the average gate
fidelity by at least three orders of magnitude even after encountering LVSPs
(teal bars).

###  III.2 Dependence of average gate fidelity on pulse parameters

Electron trajectories can be engineered for a variety of amplitudes ($x_{0}$),
frequencies ($\omega$) and pulse lengths ($T_{g}$). To study how the system
responds to these different parameters, a Ge-diffusion model based device is
used to create a well defined valley landscape, with $\kappa_{z}=$
$5\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$, and $E_{V}=103\pm 49$ . A LVSP
where the valley splitting is $\sim$$7\text{\,}\mathrm{\SIUnitSymbolMicro eV}$
is chosen along the length of the device for Rabi pulsing to demonstrate the
advantage of optimal control. The micro-magnet configuration chosen in Ref.
Künne  _et al._ (2023) is predicted to limit the maximum amplitude of
oscillations to $20\text{\,}\mathrm{nm}$ to perform EDSR.

Sweeping the values of $\omega$ for $x_{0}$ up to $20\text{\,}\mathrm{nm}$, we
calculate the average gate infidelity in the absence and presence of spin-
valley coupling, as shown in Fig. 3(a). The left panel of the figure shows the
case where $\kappa_{z}=$ $0\text{\,}\mathrm{meV}$, with optimal fidelities
obtained for $\omega\sim 560$ , corresponding to an external field of $B_{z}=$
$20\text{\,}\mathrm{mT}$. Increasing spin-valley coupling to $\kappa_{z}=$
$5\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$ (right panel) shows a shift in
resonant $\omega$ (to $\sim 562.3$ ) leading to optimal fidelities. The well
known Landau-Zener-Stückelberg-Majorana (LZSM) interference Shevchenko  _et
al._ (2010), which causes phase accumulation between excited- and ground state
when passing an anti-crossing point in energy periodically, happens in the
valley DOF. A strong spin-valley coupling results in fringes along the
frequency band. The fringes are predominant for amplitudes $<$
$10\text{\,}\mathrm{nm}$, suggesting that valley excitations are limited at
larger amplitudes.

A sweep of $x_{0}$ and $T_{g}$ is also performed, and the entire parameter
grid is optimized as mentioned in Sec. II.3 for $300$ iterations. The left
panel of Fig. 3(b) shows the parameter grid before optimization, with no
possible parameter combination yielding desired fidelities. Fringes due to
LZSM interference are visible at lower amplitudes, as noted above. The right
panel of Fig. 3(b) shows the same parameter grid after trajectory
optimization, leading to 88.68 of the parameter grid yielding infidelity below
${10}^{-3}$, highlighting the potential improvement from using optimal
control. For all amplitudes, pulse lengths $<$$42\text{\,}\mathrm{ns}$
resulted in high fidelity gates. Meanwhile, driving amplitudes
$>$$14\text{\,}\mathrm{nm}$ yielded high fidelity gates for all pulse lengths.
This suggests that even though there is a wide range of parameters where the
optimal control performs better, specific ranges of amplitudes and pulse
lengths are more desirable when optimizing the average gate infidelity.

###  III.3 Valley-model-agnostic optimal control

Valley models are differentiated by the choice of sampling the real and
imaginary part of the complex position dependent valley splitting, as
mentioned in Sec. II. Ge-diffusion and step models of valley splitting are
plausible models that can be used to simulate realistic conveyor-mode
shuttling based quantum dot devices. This section is divided into two parts.
The first part deals with the positioning of the micro-magnet along the length
of the device, and the next part deals with sampling valley profiles of $1000$
different devices. In both parts, we show that very low average gate
infidelities can be obtained, irrespective of the valley model used. In
particular, optimal control always ensures trajectories that results in high
fidelity single-qubit gates for the SpinBus architecture.

####  III.3.1 Micro-magnet position dependence

A prototypical device is simulated using both valley splitting models, with
$\kappa_{z}=$ $1\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$ and $\kappa_{z}=$
$5\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$. In a realistic scenario, Fig.
4(a) and (d) shows the valley landscape for Ge-diffusion and step model
respectively. At each point along the valley landscape, a sinusoidal electron
trajectory with amplitude $x_{0}=$ $10\text{\,}\mathrm{nm}$ and corresponding
analytical $T_{g}$ at driving frequency $\omega\sim$
$562.3\text{\,}\mathrm{MHz}$ is used to calculate average gate infidelity.
Empirically, tuning $\omega$ to account for shifts due to spin-valley
dynamics, and tuning $T_{g}$ to get rid of any over-rotation or under rotation
of the spin while performing the desired gate, could improve the average gate
fidelity in experiments. In a real experiment, calibration of $\omega$ and
$T_{g}$ could be done independently to find the sweet-spot for attaining large
gate fidelities. This process is numerically simulated using Bayesian
optimization based on Gaussian processes Garnett (2023), implemented in the
python library _scikit-optimize_ Head  _et al._ (2021). This method is faster
than the brute-force parameter sweeps for calibration, since we start the
optimization in a well defined initial interval around $\omega_{\text{Rabi}}$
and analytical $T_{g}$. Conservatively, fidelities in a real experiment might
fall between the analytical and Bayesian optimized distributions due to
limitations in precision of evaluating fidelities, for respective valley
splittings. Furthermore, the initial trajectory based on analytical $T_{g}$ is
optimized as discussed in Sec. II.3.

The results of optimization for Ge-diffusion and step model corresponding to
$\kappa_{z}=$ $1\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$ is shown in Fig.
4(b) and (e), and corresponding to $\kappa_{z}=$
$5\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$ are shown in Fig. 4(c) and (f),
with red solid line denoting the analytical infidelity, orange dotted line
denoting ($\omega,T_{g}$)-optimized infidelity and teal dash-dot line denoting
trajectory-optimized infidelity.

It is observed that the gate infidelities are valley landscape dependent,
which makes it really important to choose an appropriate operating point for
placing the micro-magnet to increase the probability of performing a high
fidelity single-qubit operation in SpinBus. As noted in Sec. III.1, larger
values of valley splitting inherently help in achieving high fidelities, even
in the presence of an unfavourable spin-valley coupling. This is evident from
the flat profile for the ($\omega,T_{g}$)-optimized infidelity at larger
valley splitting values, irrespective of the spin-valley coupling strength. It
can also be noted that the gradient of valley splitting with respect to the
position works in conjunction with the valley splitting strength to decrease
average gate fidelity (for instance, between [range-units=single, range-
phrase=–]200300 in Fig. 4(c)), due to the LZSM interference. Spin decoherence
due to LVSPs are more prevalent at larger spin-valley couplings. The
difference in the valley models chosen does not seem to change the analytical
and post-optimization fidelity characteristics.

Calibrating $\omega$ and $T_{g}$ might suffice to provide competing fidelities
for a vast majority of points along the valley landscape as shown in Fig.
4b)-(f). However, trajectory optimization always lets the average gate
fidelity converge below ${10}^{-4}$ despite encountering competing physical
effects discussed above.

####  III.3.2 Device sampling

The valley splitting models are used to generate 1000 devices of same length,
with varying valley landscape. We assume that the center of the micro-magnet
is placed exactly at the center of the $200\text{\,}\mathrm{nm}$ long device
(at $100\text{\,}\mathrm{nm}$).

_Ge-diffusion model_ : As mentioned in Sec. II, a valley profile is generated
by relative positioning of Ge atoms in the Si well. The presence of Ge atoms
alters the intervalley coupling, causing a spatially varying difference in
energy. In order to sample devices simulated using the Ge-diffusion model, a
seed is chosen to randomly place Ge atoms along the Si well. Fig 5(a) (left
panel) shows the valley splitting profile for the sampled devices. The average
valley splitting at the center taken over the $1000$ devices is $\sim 88\pm
46$ , while the average splitting over the full length of the $1000$ devices
is $\sim 88\pm 17$ . The distribution of valley splittings is skewed towards
lower splittings, with 53.9 of the samples below a splitting of $\sim$88 µeV.
Valley splittings below $33\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ are
encountered with 12.3 probability.

_Step model_ : The relative placement of miscuts along the device is tuned
using a chosen seed so as to generate different valley profiles. The
corresponding valley splitting profile is shown in Fig. 5(a) (right panel).
The average valley splitting at the center is $\sim 100\pm 44$ , while the
average splitting over thefull length of the $1000$ devices is $\sim 92\pm 15$
. The valley splittings are skewed more towards the larger values, with 56.6
of the samples above a splitting of
$\sim$$88\text{\,}\mathrm{\SIUnitSymbolMicro eV}$. Valley splittings below
$33\text{\,}\mathrm{\SIUnitSymbolMicro eV}$ are encountered with 5.6
probability, relatively less than the Ge-diffusion model. This could be
attributed to lower intervalley coupling being concentrated around the step
edges.

For both the models, we employ the same methodology as discussed in the
previous sub-section (Sec. III.3.1) to calculate analytical,
($\omega,T_{g}$)-optimized and trajectory optimized gate infidelities, with
the electron always oscillating about the center of the device for each
sample. Fig. 5(b) shows the results for Ge-diffusion (top row) and step model
(bottom row) after optimization. We observe that the analytical Rabi pulses
result in poor gate infidelities as the valley splitting decreases below
$60\text{\,}\mathrm{\SIUnitSymbolMicro eV}$, progressively getting worse for
lower valley splittings (cf. Fig. 5(b), brown bars). Calibrating $\omega$ and
$T_{g}$ can improve the fidelity for a vast majority of samples (orange bars),
but is limited by LVSPs and relatively larger spin-valley coupling values. It
can be easily seen that optimized electron trajectories performs at least
three orders of magnitude better than the analytical Rabi pulses for all
values of valley splittings.

It is evident that an important factor that plays a role in limiting the
average gate fidelity other than the valley splitting strength is the spin-
valley coupling. Gradually increasing the spin-valley coupling shifts the
attainable infidelities upwards, implying more spin decoherence, in general.
For $\kappa_{z}=$ $1\text{\times}{10}^{-6}\text{\,}\mathrm{meV}$, none of the
samples using analytical pulses converged below an average gate infidelity of
$10^{-3}$, while 779 (Ge-diffusion) and 899 (step model) out of $1000$ samples
using ($\omega,T_{g}$)-optimized pulses converged below an average gate
infidelity of ${10}^{-3}$. Trajectory shaping using optimal control for the
same leads to an average gate infidelity below ${10}^{-3}$ for all samples
irrespective of the underlying valley model or spin-valley coupling, posing a
significant advantage as opposed to the usual Rabi pulses. In conclusion,
detrimental effects of encountering LVSPs can be effectively mitigated by
optimal control.

##  IV Discussion

Conveyor-mode shuttling based architectures can be engineered with better
spin-dephasing times than the static quantum dot counterparts due to its
ability to perform larger amplitudes of electron position oscillation. We have
shown that for realistic spin decoherence rates, an order of magnitude
improvement over standard EDSR may be attainable, which makes the SpinBus
architecture a potential candidate for fault-tolerant quantum computation
Fowler  _et al._ (2012).

Oscillating the electron at larger amplitudes increases the chance of valley
excitations and positionally-dependent frequency offsets, thus hindering spin
coherence and high fidelity gates. The existence of LVSPs can severely limit
fidelity, to the extent to make these manipulation zones unusable. In this
work, we have shown that such a drastic measure is unneeded, and applying
optimal control still permits error rates below ${10}^{-3}$, enabling full use
of the SpinBus architecture. While we have focused here on model-based
control, most of the conclusions here should still hold in the case of in-situ
closed-loop control.

Meanwhile, we have shown that the valley DOF, with sufficiently large
splitting ($>$$60\text{\,}\mathrm{\SIUnitSymbolMicro eV}$) does not hamper the
spin significantly unless the spin-valley coupling is pessimistically large.
However, proper calibration to take into account position-dependent frequency
shifts is needed. The single-qubit gate fidelities after calibration can be
used to probe the unknown valley landscape along a device, to map out
potential LVSPs. This can be used to shift the choice of the initial operating
position of the electron to yield high fidelity gates without the use of
optimal control. Full optimal control enables high fidelity gates even if
there is a chance to encounter such spin-valley couplings in a realistic
device. We have modeled both the Ge-diffusion and step model scenarios in
detail, and shown that in both cases, even under pessimistic conditions,
control theory can greatly enhance device operability. It is in principle also
possible to engineer electron oscillation trajectories that are inherently
robust to the differences in valley landscapes, as mapping out the valley
landscape for large scale devices might not be practical.

Moreover, the uncertainties in measuring valley splittings in an experimental
scenario could be mitigated using closed loop control techniques, when the
trajectories could be modified to yield optimal gate fidelities real-time.
Since the average number of iterations to yield high fidelity gates for the
open loop scheme discussed in this work is around $100$, extending it to a
closed loop control scheme might increase the iterations by roughly $10$ times
Rossignolo  _et al._ (2023), which still is a break even trade-off to get high
fidelity gates, especially given fast duty cycles for these devices.

It is also important to note that there might be residual nuclear spins from
29Si even after isotopical purification, which could cause further dephasing
for the spin. Using basic driving frequency and gate time optimization in such
a scenario might not yield high fidelity gates. However these once again can
be straightforwardly improved, e.g. using dynamical decoupling Viola and Lloyd
(1998); Yoneda  _et al._ (2018), and by using the optimal control schemes
presented in this work.

##  V Conclusion

A quantum bus architecture paired with micro-magnets is promising for large
scale universal quantum computation. We have quantitatively shown that order-
of-magnitude improvement over standard EDSR is possible for shuttling based
EDSR. Nonetheless, the choice of the material stack for the device (Si/SiGe in
this work) presents additional complexity in the Hamiltonian and decay
channels that cause decoherence of the information encoded in the spins of the
electron. Higher gate fidelities are limited by the so called valley degree of
freedom arising from the conduction band minima of Si, which is further
limited by the charge noise mediated through the micro-magnets used for
single-qubit manipulation.

We show that straightforward control theoretic routines can improve the gate
fidelities by at least three orders of magnitude compared to analytical
formulas. These results are tested for two competing models for the
description of valley splittings, namely Ge diffusion and the step model. Even
for valley splittings below $15\text{\,}\mathrm{\SIUnitSymbolMicro eV}$, which
would otherwise be essentially unusable for computation, the optimized pulses
would result in 99.9 gate fidelity, thus improving controllability in tough-
to-navigate LVSPs that could be present along the length of the device.

###### Acknowledgements.

This work was supported by the Helmholtz Validation Fund project “Qruise”
(HVF-00096), and by the Germany’s Excellence Strategy – Cluster of Excellence
Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 – 390534.

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