# Dephasing and error dynamics affecting a singlet-triplet qubit during
coherent spin shuttling

Natalie D. Foster<EMAIL_ADDRESS>Sandia National Laboratories,
Albuquerque, NM 87185, USA  Jacob D. Henshaw  Sandia National Laboratories,
Albuquerque, NM 87185, USA  Martin Rudolph  Sandia National Laboratories,
Albuquerque, NM 87185, USA  Dwight R. Luhman  Sandia National Laboratories,
Albuquerque, NM 87185, USA  Ryan M. Jock  Sandia National Laboratories,
Albuquerque, NM 87185, USA

###### Abstract

Quantum information transport over micron to millimeter scale distances is
critical for the operation of practical quantum processors based on spin
qubits. One method of achieving a long-range interaction is by coherent
electron spin shuttling through an array of silicon quantum dots. In order to
execute many shuttling operations with high fidelity, it is essential to
understand the dynamics of qubit dephasing and relaxation during the shuttling
process in order to mitigate them. However, errors arising after many repeated
shuttles are not yet well documented. Here, we probe decay dynamics
contributing to dephasing and relaxation of a singlet-triplet qubit during
coherent spin shuttling over many $N$ repeated shuttle operations. We find
that losses are dominated by magnetic dephasing for small $N<10^{3}$ and by
incoherent shuttle errors for large $N>10^{3}$. Additionally, we estimate
shuttle error rates below $1\times 10^{-4}$ out to at least $N=10^{3}$,
representing an encouraging figure for future implementations of spin
shuttling to entangle distant qubits.

##  I Introduction

Spin qubits are a compelling candidate for the foundation of a quantum
register network, having demonstrated single and two-qubit gate fidelities
well over 99% Yoneda2018 ; Mills2022 . By reducing the spinful 29Si nuclear
background in silicon, isotopic enrichment enables coherent spin qubits to be
hosted in foundry-compatible Zwerver2022 ; Neyens2024  gate-defined quantum
dots (QDs) vanderWiel2006 ; Burkard2023 , streamlining integration with
existing and rapidly advancing industrial silicon technology. Since spin qubit
operations are carried out using short-range interactions often involving
nearest neighbors Noiri2022 ; Watson2018 ; Xue2022 ; Sigillito2019 , a
scalable quantum computing architecture relying on long-range quantum
information transfer using spin qubits could present a challenge. One solution
is to build a remote qubit registry that facilitates information exchange over
broad physical scales, leaving qubit operations to be carried out at dedicated
local nodes VanMeter2013 ; Vandersypen2017 .

Linking quantum information between qubit nodes can be achieved in multiple
ways. One example is by interfacing distant stationary qubit sites through an
intermediate messenger, such as through virtual microwave photons to mediate
remote spin-spin interactions in quantum transducers Mi2018 ; Borjans2020 ;
Harvey-Collard2022 . Another approach is to physically move the spin qubit;
for example, by using surface acoustic waves to form a moving QD Takada2019 ;
Jadot2021 , or by shuttling the spin controllably through an array of gate-
defined electrostatic potential traps Flentje2017 ; Fujita2017 ; vanRiggelen-
Doelman2023 ; Langrock2023 . In the latter, the ability to perform many
repeated shuttling operations with a low error rate is key to maintaining
long-range qubit entanglement. A firm understanding of dephasing and
relaxation sources during shuttling is therefore required. At present, these
dynamics are not well quantified, especially out to the large number of
shuttles needed to perform long-range interactions. For a spin qubit device
composed of millions of qubits, for example, arranged on a crossbar array
Borsoi2024 , thousands to tens of thousands of shuttling operations will need
to be performed to link sites on either end of the processor.

In this work, we categorize the main error sources and dynamics during
electron spin shuttling by observing a singlet-triplet qubit over many
repeated coherent shuttles of a single electron spin in a gate-defined Si/SiGe
triple QD device. We focus first on expected shuttling-independent loss
mechanisms – inhomogeneous dephasing and spin relaxation – and use this
baseline to classify the nature of errors dependent on the number of shuttles
($N$). From Ramsey decay measurements carried out on the stationary qubit and
the qubit during shuttling, we attribute the dominant error source occurring
during shuttling for $N<10^{3}$ to inhomogeneous dephasing arising from
residual nuclear spins. We corroborate this finding by using dynamical
decoupling to reduce magnetic noise and show it is independent of $N$, while
also discovering the presence of nontrivial noise dynamics. From relaxation
measurements, we define a bounding timescale expected of incoherent error
processes in the qubit. We find, after implementing many repeated shuttling
operations, that shuttling errors are incoherent but don’t contribute
appreciably to losses until $N>10^{3}$, meaning that errors due to shuttling
are small. Understanding the various noise mechanisms encountered during
shuttling can aid the development of targeted mitigation strategies that will
enable the completion of a large number of shuttles, allowing an opportunity
for linking localized Morton2008 ; Pla2013 ; Witzel2022 , and even
deterministic Jakob2022 , qubit nodes on a sparse qubit array.

###  I.1 Device overview and operation

Fig. 1a shows a schematic of the gate-defined triple QD device used in this
work. Details of the device structure can be found in the Methods section. The
electrochemical potential and electron occupation $n_{\mathrm{QD}i}$ at each
QD$i$ are primarily controlled by voltages on plunger gates P$i$, where
$i=0,1,2$ is the QD index. We denote the total charge occupation across the
triple QD array by $(n_{\mathrm{QD0}},n_{\mathrm{QD1}},n_{\mathrm{QD2}})$. The
tunneling rate from the electron reservoir is controlled by a voltage applied
to the barrier gate B0 and the tunnel coupling between adjacent QDs is
controlled by barrier gates B1 and B2 (Fig. 1b).

We operate the device with four electrons in the QD chain, where two low-
energy electrons under P0 form a filled-shell singlet and do not contribute
significantly to the spin dynamics of the system Higginbotham2014 ; Harvey-
Collard2017 . Fig. 1c shows an energy diagram of the remaining two-spin system
illustrating the relationship between the singlet ($\ket{S}$) and triplet
($\ket{T_{0}},\leavevmode\nobreak\ \ket{T_{+}},\leavevmode\nobreak\
\ket{T_{-}}$) states in a small magnetic field. We employ a singlet-triplet
($S$-$T_{0}$) qubit Petta2005 , where the qubit is encoded in the $m=0$
subspace of the 4-level system defined by linear combinations of the
$\ket{\uparrow\downarrow}$ and $\ket{\downarrow\uparrow}$ states. Its logical
states are defined as the eigenstates of the system in the limit of large
exchange:

$\displaystyle\ket{0}=\ket{S}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})$
(1)
$\displaystyle\ket{1}=\ket{T_{0}}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})$

A schematic of the pulse sequence used to initialize and measure the qubit is
depicted in Fig. 1e, where we use an enhanced latching mechanism Harvey-
Collard2018  of spin-to-charge conversion through Pauli spin blockade (PSB) to
measure the qubit spin state Johnson2005 ; Lai2011 ; Zhao2019 ; Jang2020 .
Two-axis control of the qubit is achieved through baseband voltage pulses to
modify barrier and QD chemical potentials and toggle between control regimes
dominated by the exchange interaction, $J$, and the difference in electron
Zeeman energy splitting between electrons in two QDs, $\Delta E_{\mathrm{Z}}$
(Fig. 1d). We use voltage pulses applied to the B1 barrier gate and adjacent
P0 and P1 QD gates at the exchange symmetric operating point Martins2016 ;
Reed2016  to achieve rotations between $\ket{\uparrow\downarrow}$ and
$\ket{\downarrow\uparrow}$. When the electrons are well isolated and exchange
is turned off, $\ket{S}$ to $\ket{T_{0}}$ rotations are driven by Jock2022

$\Delta E_{\mathrm{Z}}=\Delta g\mu_{\mathrm{B}}B,$ (2)

where $g$ is the $g$-factor, $\mu_{\mathrm{B}}$ is the Bohr magneton, and $B$
is the magnetic field. Differences in local magnetic fields or local
$g$-factors both contribute to $\Delta E_{\mathrm{Z}}$. The unique $B$ at each
QD arises from external magnetic field gradients, or from the presence of
residual spinful nuclei forming a random but slowly-varying Overhauser field
Koppens2005 ; Coish2008 . The difference in $g$-factor at each QD comes from
variations in interface disorder across the Si/SiGe interface giving rise to
local spin-orbit interactions Prada2011 ; Jiang2012 .

##  II Results

###  II.1 Coherent spin transfer

Two established approaches for spin shuttling Langrock2023  are conveyor mode
Mills22019 ; Seidler2022 ; Xue2023 ; Struck2024  and bucket brigade mode
Flentje2017 ; Fujita2017 ; Noiri2022 ; vanRiggelen-Doelman2023 ; Zwerver2023 .
Shuttling in this work is accomplished by a barrier pulse-assisted bucket
brigade mode, in which we leverage individual gate voltage control to tailor
the electrostatic potential landscape and move the spin between QDs. We
prepare a $\ket{S}$ state and transfer one electron of the entangled spin pair
from QD1 to QD2 (charge states (3,1,0) and (3,0,1), respectively) by detuning
the voltages $V_{\mathrm{P1}}$ and $V_{\mathrm{P2}}$ (ramping in opposite
directions), while simultaneously pulsing $V_{\mathrm{B2}}$ to lower the
potential barrier and facilitate adiabatic transfer (Fig. 2a) (see discussion
on Supplementary Fig. 1 for more details). We hold $V_{\mathrm{B1}}$ fixed to
maintain negligible residual $J$ with the electron in QD0 and ensure rotations
about the Bloch sphere are facilitated by the dominant $\Delta E_{\mathrm{Z}}$
interaction.

We first demonstrate spin transfer from one QD to another by examining qubit
rotations in the region near the (3,1,0)-(3,0,1) charge transition. We apply
an external magnetic field of $B_{0}=50$ mT in the plane of the Si quantum
well to intentionally drive strong $S$-$T_{0}$ rotations in both charge
regions, showing the spin is coherently transferred. Fig. 2b shows singlet
return probability following a 6 $\mu$s wait time at different QD1-QD2
detunings. This plot has four regions of interest describing different charge
configurations. The bands in (3,1,0) and (3,0,1) describe differences in
acquired singlet phase within each charge configuration due to changes in $g$
modifying $\Delta E_{\mathrm{Z}}$ as QD1 and QD2 are shaped by applied
voltages, and the local interface disorder sampled by the electron varies
Liu2021 . The charge transition between the (3,1,0) and (3,0,1) appears as a
diagonal strip of constant $\sim$50% singlet return probability with no
acquired phase accumulation visible, indicating a fully dephased qubit. A
shift in coherent singlet return on either side of the transition indicates
the spin is transferred from one isolated QD to the other (see discussion on
Supplementary Fig. 2 for more details). The low singlet return probability and
absence of acquired phase bands in (3,1,1) ((3,0,0)) is due to the addition
(subtraction) of an electron to the system which rapidly destroys qubit
coherence. Altogether, these four delineated regions map out the charge space
where spin shuttling operations will be performed and show that the entangled
electron spin may be coherently transferred between QD1 and QD2.

###  II.2 Spin coherence and sources of dephasing

To demonstrate spin coherence is preserved at each shuttle step and to
quantify noise dynamics that drive dephasing independent of shuttling, we
perform a Ramsey decay experiment by preparing a $\ket{S}$ state and letting
it evolve under $\Delta E_{\mathrm{Z}}$. We measure the singlet return
probability over time, averaged over 1 hour total to ensure the ergodic limit
was reached Madzik2020  (Supplementary Fig. 3). Fig. 2c shows the singlet
return at different locations on the charge stability map in Fig. 2b indicated
as colored circles corresponding to (3,1,0), (3,0,1), and the charge
transition.

We first consider dephasing in the case of negligible external magnetic field
($B_{0}\approx 0$ mT) such that dephasing is expected to be dominated by
fluctuations in the local Overhauser field Petta2005 , where no $S$-$T_{0}$
rotations are observed. We fit the data to a decay profile of the form

$f_{T_{2}^{*}}(t)=Ae^{-\left(t/T_{2}^{*}\right)^{\alpha}}+y_{0},$ (3)

where $A$ is the amplitude, $y_{0}$ is the vertical offset, $T_{2}^{*}$ is the
inhomogeneous dephasing time, and $\alpha$ is the decay exponent. The
resulting dephasing times and fit uncertainties are $T_{2}^{*}(3,1,0)=4.60\pm
0.02\leavevmode\nobreak\ \mu$s, $T_{2}^{*}(3,0,1)=3.86\pm
0.02\leavevmode\nobreak\ \mu$s, and $T_{2}^{*}(\mathrm{transition})=5.12\pm
0.03\leavevmode\nobreak\ \mu$s, and are consistent with previously measured
quantities for QDs in 800 ppm 29Si Eng2015 ; Jock2018 ; Neyens2024 . The
variation among $T_{2}^{*}$s suggest local sources of dephasing affect the
spin in each QD, including the wavefunctions of nearby 29Si and 73Ge. We
interpret the longer $T_{2}^{*}(\mathrm{transition})$ as the result of a
process like motional narrowing – effectively averaging out magnetic noise
acquired from the different nuclear distributions when the spin is hybridized
between QD1 and QD2 Langrock2023 . The resulting decay exponent fits,
$\alpha(3,1,0)=1.92\pm 0.02$, $\alpha(3,0,1)=2.16\pm 0.02$, and
$\alpha(\mathrm{transition})=2.12\pm 0.02$, resemble Gaussian profiles as
expected of inhomogeneous nuclear spin dephasing Petta2005 .

Next, we apply a small external field of $B_{0}=50$ mT to amplify and study
the effect of spin-orbit-driven charge noise acting on $\Delta
E_{\mathrm{Z}}$. Clear $S$-$T_{0}$ oscillations are observed and fit to a
decaying sine function of the form

$f_{T_{2}^{*}}(t)=Ae^{-\left(t/T_{2}^{*}\right)^{\alpha}}\sin\left(2\pi
ft\right)+y_{0},$ (4)

where $f$ is the $S$-$T_{0}$ rotation frequency. The resulting dephasing times
are $T_{2}^{*}(3,1,0)=3.82\pm 0.12\leavevmode\nobreak\ \mu$s,
$T_{2}^{*}(3,0,1)=4.09\pm 0.03\leavevmode\nobreak\ \mu$s, and
$T_{2}^{*}(\mathrm{transition})=1.08\pm 0.01\leavevmode\nobreak\ \mu$s. We
attribute the fast decay observed at the charge transition to the
hybridization of the electron wavefunction split between QD1 and QD2
Yoneda2021 , which is sensitive to charge noise. Charge noise arises from
rapid fluctuations in spin-orbit interactions at the Si/SiGe interface that
are intensified in an applied external magnetic field.

Additionally, we observe changes to $\alpha$ that imply changes to the noise
spectrum. The resulting decay exponent fits are $\alpha(3,1,0)=0.90\pm 0.06$,
$\alpha(3,0,1)=2.12\pm 0.06$, and $\alpha(\mathrm{transition})=1.14\pm 0.02$,
ranging from exponential to Gaussian forms. Since it is the noise spectrum and
mechanisms that dictate the observed decay profile Stano2022 , the spread in
$\alpha$ suggests that different localized noise dynamics, in addition to
residual nuclear spins, contribute to dephasing in each QD. One possible
source known to exhibit non-Gaussian decay behavior is random telegraph noise
originating from charge traps Culcer2009 , where the observed charge noise is
often attributed to two-level systems (TLSs) that are distributed nonuniformly
across the Si/SiGe interface Connors2019 . This is consistent with our
assessment that charge noise dominates the fast
$T_{2}^{*}(\mathrm{transition})$, and suggests an increased susceptibility to
charge noise in QD1 compared to QD2, since $\alpha(3,1,0)$ is exponential and
$\alpha(3,0,1)$ is Gaussian. Overall, we observe qubit dephasing to be
primarily driven by fluctuations in the Overhauser fields generated by
residual nuclear spins at 0 mT, and a combination of nuclear spins and charge
noise at 50 mT.

###  II.3 Repeated spin shuttles

To understand the role of shuttle-induced errors, independent of intrinsic
noise, we perform repeated spin shuttling operations to investigate errors
from shuttling and observe the singlet return probability. We implement
repetitions of the procedure outlined in the previous section to transfer an
electron back and forth between QD1 and QD2. The voltage pulses are structured
such that the shuttled electron spends an equal amount of time in each QD
during shuttling ($2t_{\mathrm{wait}}$), and the ramp time
($t_{\mathrm{ramp}}$) is fixed at 10 ns between charge configurations to
ensure adiabatic charge transfer. A single shuttle is defined as one transfer
of an electron between QD1 and QD2 (from either of (3,1,0) or (3,0,1) to the
other), where the total time spent shuttling $N$ times is
$t_{\mathrm{shuttle}}=N(2t_{\mathrm{wait}}+t_{\mathrm{ramp}})$ (Fig. 3a).

To investigate dephasing during shuttling, we vary $t_{\mathrm{wait}}$ for a
series of fixed $N$ carried out within $t_{\mathrm{shuttle}}$ (Fig. 3c), with
$B_{0}\approx 0$ mT to minimize the charge noise-induced field-dependent noise
dynamics on $\Delta E_{\mathrm{Z}}$ discussed above. We observe a single decay
profile in singlet return, independent of $N$, indicating the decay is
primarily time-dependent. A decay fit of the form in Eq. 3 to the combined
data reveals $T_{2}^{*}=5.07\pm 0.07\leavevmode\nobreak\ \mu$s, which is
similar to $T_{2}^{*}(\mathrm{transition})$ measured in the absence of
shuttling and $B_{0}$ (Fig. 2c), and indicates the spin samples similar noise
environments when shuttled and hybridized. This observation suggests the error
introduced by the act of shuttling is low. The decay profile fits to a near-
Gaussian with $\alpha=1.82\pm 0.07$, which points to the dominant dephasing
mechanism being nuclear spin fluctuations.

Next, we employ a standard spin echo technique Jock2018  to reduce the effect
of slow magnetic noise and extend the qubit dephasing time. This allows more
shuttle operations to be performed within the qubit coherence time, to further
highlight the role of shuttle-induced error dynamics. For this experiment,
$B_{0}\approx 0$ mT to minimize field-dependent effects on $\Delta
E_{\mathrm{Z}}$. We prepare the $\ket{S}$ state and allow the two electron
spins to evolve while one is stationary in QD0 and the other is repeatedly
shuttled between QD1 and QD2 $N/2$ times. A calibrated $J$ pulse on B1 applies
a $\pi_{z}$ operation to swap the two spins (Fig. 3b), followed by $N/2$
shuttles such that both spins sample the same magnetic environment for an
equal amount of time, negating any acquired phase difference due to nuclear
spin fluctuations slower than $1/2t_{\mathrm{shuttle}}$. Fig. 3d shows the
singlet return probability after implementing the spin echo sequence during
shuttling for a range of $N$ completed within $t_{\mathrm{shuttle}}$. We
observe a single decay profile independent of $N$ and fit the data to a
generic decay function of the form

$f_{T_{2}^{\mathrm{Hahn}}}(t)=Ae^{-\left(t/T_{2}^{\mathrm{Hahn}}\right)^{\alpha}}+y_{0},$
(5)

similar to Eq. 3. The fit yields $T_{2}^{\mathrm{Hahn}}=84.48\pm
7.82\leavevmode\nobreak\ \mu$s which is reasonably consistent with previously
measured values in Si/SiGe heterostructures Kawakami2014 ; Kerckhoff2021 , and
indicates sources of noise contributing to dephasing have been reduced. The
fit converges with an exponent of $\alpha=0.64\pm 0.06$, which deviates from
$\alpha$ commonly observed in the range of $1\lesssim\alpha\lesssim 2$
Petta2005 ; Kawakami2014 ; Veldhorst2014 ; Kobayashi2021 , suggesting a
different noise spectrum is introduced. In other systems, sub-exponential
decoherence behavior has been theorized to originate from 2D electron surfaces
Witzel2012 , and has recently been predicted in 2D electron spin baths in NV-
centers in diamond Schatzle2024 . The sub-exponential decay behavior observed
here points to nontrivial noise dynamics of this nature affecting the qubit.

Next, we investigate the role of spin relaxation during shuttling since spin-
flip mechanisms are known error sources in Si QDs Yang2013 . We prepare an
eigenstate $\ket{\uparrow\downarrow}_{(3,1,0)}$ of the $\Delta E_{\mathrm{Z}}$
interaction and monitor return of this state while repeatedly shuttling
between QD1 and QD2. We used a small applied field of $B_{0}=10$ mT to avoid
mixing with $\ket{T_{-}}$ during the (3,1,0) charge state preparation. Fig. 3e
shows singlet return probability of the shuttled $\ket{\uparrow\downarrow}$
state for a range of $N$ completed within $t_{\mathrm{shuttle}}$. We observe a
single decay profile independent of $N$ and fit the data to an exponential
decay function of the form

$f_{T_{1}}(t)=Ae^{-t/T_{1}}+y_{0}$ (6)

with a range of assumptions placed on $y_{0}$, the singlet return saturation,
since it is not explicitly measured. The fit results in $T_{1}=6.22\pm 0.71$
ms, where the uncertainty represents fit variation based on $y_{0}$, and lies
within the broad range of $T_{1}$ measured for electrons confined to Si QDs,
typically falling between milliseconds and seconds Simmons22011 ; Yang2013 ;
Petit2018 ; Eng2015 . A variety of mechanisms generally stemming from spin-
orbit and spin-valley interactions contribute to spin relaxation in Si QDs
Tahan2014 ; Borjans2019 . Due to the natural disorder and inversion asymmetry
at the Si/SiGe interface, the small in-plane $B_{0}$, as well as shuttling
itself, give rise to spin-orbit effects Jock2018  that can induce spin flip
relaxation over time. At $B_{0}=10$ mT, we expect a small but non-negligible
influence from the spin-valley hotspot measured near 190 mT in QD0 (see
discussion on Supplementary Fig. 4), which could drive a faster relaxation
rate ($T_{1}^{-1}$) Yang2013 . Very few references for $T_{1}$ in the low-
field limit exist; however, the observation that $T_{1}>T_{2}^{\mathrm{Hahn}}$
in our device shows that incoherent errors due to spin-flip processes during
shuttling are relatively small compared to coherent errors arising from
magnetic noise.

From the data presented in Figs. 3c, d, and e, we anticipate the shuttling
error to be low since the time-dependent singlet return decays in each panel
are nearly indistinguishable among different $N$. To assess the time-
independent error due to the shuttling operation, we fix
$t_{\mathrm{shuttle}}$ while varying $N$ such that the effect of time-
dependent dephasing and errors is constant with $N$. Figs. 3f, g, and h show
the measured singlet return probability at three fixed $t_{\mathrm{shuttle}}$
corresponding to the shuttling experiments shown in the insets of Figs. 3c, d,
and e, respectively. The error bars represent standard deviations of the mean
of singlet return spanning repeated measurements totaling at least one hour.
In all cases, the decay in singlet return is undetectable within error bars,
and suggests shuttle errors are negligible out to at least $N=10^{3}$ (Fig.
3g). We make an attempt to quantify the worst-case estimate of shuttling error
by fitting a simple linear function from the top of the first error bar to the
bottom of the last error bar and extracting the slope of the line (not shown),
which bounds the error per shuttle below $1\times 10^{-4}$, an improvement
from recently reported shuttling errors vanRiggelen-Doelman2023 ; Struck2024 .

Due to the low shuttling error, many repeated $N$ are required to identify and
elucidate the nature of shuttle-induced losses. To do so we extend the
previously discussed spin echo shuttling experiment out to the maximum $N$
possible within our instrument bandwidth capabilities. We compose a shuttling
experiment at $B_{0}\approx 0$ mT to reduce field-dependent noise, with the
minimum possible $2t_{\mathrm{wait}}=4$ ns and increase the repeated shuttles
out to $N=4\times 10^{5}$, spanning a total time of $t_{\mathrm{shuttle}}=5.6$
ms, where $t_{\mathrm{shuttle}}(N)=14N$ ns. The singlet return probability
over $N$, and equivalent $t_{\mathrm{shuttle}}$, is shown in Fig. 4. We
consider the observed decay in the context of estimated
$T_{2}^{\mathrm{Hahn}}$ and $T_{1}$ dynamics in this device, which are
presented as shaded colored regions and are scaled by a multiplicative factor
accounting for SPAM variation, serving as a guide to the eye. For a small
number of shuttles $N<10^{3}$, the $T_{2}^{\mathrm{Hahn}}$ decay profile
agrees with the data and implies the decay is dominated by the nontrivial
spectral noise acting on $\Delta E_{\mathrm{Z}}$. For a large number of
shuttles $N>10^{3}$, the decay deviates from this behavior and appears to be
dominated by a different mechanism. The observation that the singlet return
saturates below 50% indicates that the shuttling error is due to an incoherent
spin relaxation process. The process causes decay on a timescale faster than
the measured $T_{1}$ (for $N<10^{3}$), indicating a shuttle-induced, or
$N$-dependent mechanism dominating spin relaxation when shuttling a large
number of times ($N>10^{3}$).

To characterize the nature of the shuttling error as a function of the number
of shuttles, we fit to the data a phenomenological function of the form

$f_{N^{*}}(N)=Ae^{-(N/N^{*})^{\alpha}}+y_{0},$ (7)

which defines the characteristic decay in units of $N$ ($N^{*}$). The fit
results in a sub-exponential $\alpha=0.41\pm 0.03$, which suggests the
presence of noise mechanisms that deviate from both a typical $1/f$-like noise
spectrum observed for simple dephasing Petta2005 ; Connors2022  and a simple
exponential decay expected of relaxation Simmons2011 ; Petit2018 . The
observation that the function fits poorly for smaller $N<10^{3}$ suggests the
influence of shuttle errors do not become apparent until after the completion
of $N\sim 10^{3}$ shuttles. From the fit we also find $N^{*}=(1.92\pm
0.32)\times 10^{4}$ shuttles can be completed within a shuttle error-limited
characteristic decay period, an order of magnitude more than recently
demonstrated with shuttled spins Zwerver2023 ; vanRiggelen-Doelman2023 .
Although a single decay profile is insufficient to describe the losses
observed during shuttling, it points to the complexity of competing noise and
error mechanisms at play during shuttling as $N$ increases.

Shuttle experiment | Decay fit | Lifetime ($\tau$) | Exponent ($\alpha$) | Error per shuttle ($p$)
---|---|---|---|---
Ramsey | Gaussian |  $T_{2}^{*}=5.07\pm 0.07\leavevmode\nobreak\ \mu$s | $1.82\pm 0.07$ | $2.2\times 10^{-5}$
Relaxation | Exponential |  $T_{1}=6.22\pm 0.71$ ms | 1 (fixed) | $2.3\times 10^{-6}$
Table 1: Shuttle errors based on decay fit calculated from the time it takes
to complete one shuttle ($p=1-e^{-(14\leavevmode\nobreak\
\mathrm{ns}/\tau)^{\alpha}}$)

##  III Discussion

Understanding sources of error during shuttling is essential to the
development of a successful spin shuttling platform used to entangle remote
qubits. In this work, we have built on previous efforts exploring bucket
brigade mode shuttling in Si Yoneda2021 ; Noiri2022  and in Ge vanRiggelen-
Doelman2023 . Using a enhanced barrier pulse-assisted approach, we have
demonstrated coherent spin shuttling in the limit of a large number of
shuttles, with the goal of categorizing the different dephasing and relaxation
mechanisms affecting the qubit while shuttling. Independent of shuttling, we
identify magnetic noise from Overhauser field fluctuations as the dominant
source of dephasing in the absence of external fields ($B_{0}\approx 0$ mT)
and a combination of nuclear bath noise and charge noise originating from TLSs
at the Si/SiGe interface at finite field ($B_{0}=50$ mT). When the spin is
repeatedly shuttled, we observe little change in this decay at $B_{0}\approx
0$ mT, indicating that shuttling operations negligibly contribute to losses.
Using a standard spin echo technique, we show coherence time during shuttling
may be extended owing to the reduction of magnetic noise on $\Delta
E_{\mathrm{Z}}$ and is limited by a different noise spectrum indicated by a
subexponential $T_{2}^{\mathrm{Hahn}}$ decay profile. Residual unfiltered
magnetic noise on $T_{2}^{\mathrm{Hahn}}$ timescales is determined to be the
limiting time-dependent contribution since $T_{2}^{\mathrm{Hahn}}<T_{1}$.
Additionally, we find that $T_{2}^{*}$, $T_{2}^{\mathrm{Hahn}}$, and $T_{1}$
are $N$-independent in the regime examined in this work and do not contribute
appreciably to shuttling error.

By executing shuttle operations out to $N=4\times 10^{5}$, we are able to
identify a shuttling-induced incoherent error which contributes significantly
only after many shuttle operations. In the small $N$ ($<10^{3}$) regime,
shuttling errors are negligible and decay is primarily limited by residual
magnetic dephasing on $T_{2}^{\mathrm{Hahn}}$ timescales. In the large $N$
($>10^{3}$) regime, a shuttle-induced, or $N$-dependent mechanism driving spin
relaxation causes decay on a timescale faster than the measured $T_{1}$ (for
$N<10^{3}$), which dominates and limits decay during the shuttling sequence.
In the absence of temporal dephasing and relaxation (Figs. 3f., g., h.), we
made a rudimentary yet encouraging shuttle error estimate of $1\times
10^{-4}$. A more thorough assessment is required to elucidate the details of
this $N$-dependent process. Future work is required to improve
characterization of the noise spectrum affecting the qubit during shuttling in
order to extend shuttling capability, which can be done, for example, by
spectrally targeting coherent noise on the gates Connors2022 .

Lastly, we make estimates of the error per shuttle based on decay assumptions
for 14 ns shuttling operations, summarized in Table 1. Of the dynamics
described by the three error regimes evaluated in this work, error estimates
based on the $T_{2}^{*}$ and $T_{1}$ decays represent the more practical
expectation for building up a long-range spin shuttling platform. If
fluctuating Overhauser fields are the only noise mechanism and decay is
limited by $T_{2}^{*}$, we extrapolate the shuttling error to $2.2\times
10^{-5}$. In the case that spin-flip relaxation on the $T_{1}$ timescale
limits decay, the error is estimated at $2.3\times 10^{-6}$. These results
highlight the potential for low-error shuttling as a promising pathway to
enabling distant spin entanglement in quantum computing platforms.

##  IV Methods

###  IV.1 Device details

The device was fabricated by Intel Corporation and is based on a Si/Si0.7Ge0.3
heterostructure Neyens2024 , where the quantum well is composed of
isotopically-purified 28Si (800 ppm 29Si). Gate-defined QDs are formed by
molding the local electrostatic potential with voltages applied to the
surrounding gates. A reservoir of electrons is hosted under the accumulation
gate AC0 from which the QDs are populated. Changes in charge occupation are
sensed by a single electron transistor (SET) located at a dedicated QD on the
top half of the device opposite the qubit QDs and isolated by a center
screening gate. The dot pitch is $\sim$120 nm and the well width is $\sim$5
nm.

###  IV.2 Experimental setup

We used a closed-loop dry cryostat Bluefors LD400 dilution refrigerator with a
base temperature of 7 mK. The electron temperature was approximately 250 mK.
We used a heterojunction bipolar transistor (HBT) operating at -1.056 V
emitter bias on the readout SET line to perform cryogenic preamplification
Curry2019 . DC voltage to the gates was supplied by a QDevil QDAC II and fast
AC pulses were generated by a Keysight M3201 AWG with a 500 MHz sampling rate.
DC and fast voltage pulses are combined through a cryogenic RC bias tee at the
PCB. Each individual spin state measurement is acquired over an average of 100
shots. We used one axis of a superconducting vector magnet aligned to the
[110] crystal direction in the plane of the device, perpendicular to the
direction of shuttling.

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##  VI Acknowledgments

We gratefully acknowledge Intel Corp. for providing devices and Matthew Curry,
Lester Lampert and Nathan Bishop for helpful discussions on devices operation
and performance. We thank Charlotte I. Evans for offering advice on
measurement acquisition software usage. We also thank Wayne M. Witzel and N.
Toby Jacobson for fruitful conversations regarding the observed decay
dynamics. Sandia National Laboratories is a multi-mission laboratory managed
and operated by National Technology & Engineering Solutions of Sandia, LLC
(NTESS), a wholly owned subsidiary of Honeywell International Inc., for the
U.S. Department of Energy’s National Nuclear Security Administration
(DOE/NNSA) under contract DE-NA0003525. This written work is authored by an
employee of NTESS. The employee, not NTESS, owns the right, title and interest
in and to the written work and is responsible for its contents. Any subjective
views or opinions that might be expressed in the written work do not
necessarily represent the views of the U.S. Government. The publisher
acknowledges that the U.S. Government retains a non-exclusive, paid-up,
irrevocable, world-wide license to publish or reproduce the published form of
this written work or allow others to do so, for U.S. Government purposes. The
DOE will provide public access to results of federally sponsored research in
accordance with the DOE Public Access Plan.

##  VII Author contributions

N.D.F. and R. M. J. designed the experiments. N. D. F. performed the
experiments and analyzed the data. N. D. F., J. D. H., and R. M. J. discussed
the results. J. D. H. and M. R. contributed to the measurement setup and
performed initial shuttling demonstrations. N. D. F. wrote the manuscript with
input from all co-authors. R. M. J. and D. R. L. supervised the project.

##  VIII Supplementary Information

fit parameter | (3,1,0) at 0 mT | (3,0,1) at 0 mT | transition at 0 mT | (3,1,0) at 50 mT | (3,0,1) at 50 mT | transition at 50 mT
---|---|---|---|---|---|---
$A$ ($\ket{S}$ %) | $0.325\pm 0.002$ | $0.354\pm 0.002$ | $0.318\pm 0.002$ | $0.342\pm 0.009$ | $0.330\pm 0.003$ | $0.321\pm 0.003$
$T_{2}^{*}$ ($\mu$s) | $4.60\pm 0.03$ | $3.86\pm 0.01$ | $5.12\pm 0.03$ | $3.82\pm 0.12$ | $4.09\pm 0.03$ | $1.08\pm 0.01$
$\alpha$ (a. u.) | $1.92\pm 0.03$ | $2.16\pm 0.03$ | $2.10\pm 0.03$ | $0.90\pm 0.06$ | $2.12\pm 0.06$ | $1.14\pm 0.02$
$f$ (MHz) | - | - | - | $2.4031\pm 0.0008$ | $2.0356\pm 0.0007$ | $2.3516\pm 0.0010$
$y_{0}$ ($\ket{S}$ %) | $0.374\pm 0.001$ | $0.408\pm 0.001$ | $0.399\pm 0.001$ | $0.473\pm 0.001$ | $0.481\pm 0.001$ | $0.4640\pm 0.0001$
Table 2: Fit parameters and uncertainties for Ramsey experiment shown in Fig.
2c.

###  VIII.1 Barrier pulsing

For barrier-assisted bucket brigade shuttling we pulse the B2 barrier gate to
increase the coupling between QD1 and QD2 and ensure adiabatic spin transfer.
Supplementary Fig. 1 shows a series of singlet return probability raster scans
in the P1-P2 voltage space (left column), accompanied by linear detuning ramps
of $V_{\mathrm{P1}}$ and $V_{\mathrm{P2}}$ (ramped in opposite directions)
from (3,1,0) to (3,0,1) over time (right column). Nominally
$V_{\mathrm{B2}}^{0}=0.81$ V before the transfer, and dB2$\leavevmode\nobreak\
=V_{\mathrm{B2}}-V_{\mathrm{B2}}^{0}$ V. For high barrier (small dB2, first
row), the electron spin is transferred to (3,0,1) with low fidelity evidenced
by the decrease in oscillation amplitude. The appearance of multiple
$S$-$T_{0}$ oscillation frequencies in the first three rows with dB2 = 0.100,
0.125, 0.150 V (panels f.-h.) suggest the possibility of leakage into excited
valley-orbit states Yang2012  during transfer. For dB2 = 0.175, 0.200 V in the
last two rows, the $\ket{S}$ return probability is similar for both charge
configurations, indicating the electron spin is transferred with minimal
coherence loss while preserving the individual character of the QDs. The
quickly dephasing singlet at the charge transition in panels f.-j. suggests
the electron is strongly hybridized in both QDs. The criteria used to select
dB2 for the experiments were singlet visibility preservation during the
transfer and clear distinction of the charge transition, which are optimal in
the case of dB2 = 0.200 V.

###  VIII.2 Characterizing $\Delta g$ and the interdot charge transition

We expect the rotation frequency $f$ to increase linearly with $B_{0}$ at a
rate $\Delta g_{i,j}\mu_{\mathrm{B}}/h$ defined for each pair of QD$i$ and
QD$j$, where $h$ is the Planck constant from the expression $E=hf$. We
measured the singlet return probability as a function of wait time in each
charge configuration (3,1,0) and (3,0,1), while an in-plane $B_{0}$ oriented
along the [110] crystallographic axis was ramped from 0 to 300 mT.
Supplementary Fig. 4 shows the raw data and normalized fast Fourier
transforms, demonstrating $S$-$T_{0}$ rotation frequency increasing linearly
with $B_{0}$ at lower fields for both QD0-QD1 and QD0-QD2 pairs, as expected.
Linear fits to the data result in distinct slopes of 42.47 $\pm$ 0.21 MHz/T
for QD0-QD1, and 36.26 $\pm$ 0.71 MHz/T for QD0-QD2 from which we estimated
$\Delta g$-factors of $|\Delta g_{0,1}|=0.0030$ and $|\Delta g_{0,2}|=0.0026$.
The observation of rapid $S$-$T_{0}$ rotations and divergence in frequency for
both charge configurations near $B_{0}=$ 190 mT suggests the presence of
strong spin-valley coupling in QD0, the dot common to both configurations
Jock2022 . The appearance of a second divergence near $B_{0}=0.3$ T could be
the emergence of a weakly coupled higher-energy valley state that is more
pronounced in QD1 than in QD2. Nonetheless, for the low fields used in this
work ($B_{0}=0$ to 50 mT), we expect the influence of these unexplored
features to be minimal.

We analyze the nature of the interdot charge transition in closer detail in
Supplementary Fig. 2 by transferring the prepared qubit state between QD1 and
QD2 and observing the singlet return probability over time. Supplementary Fig.
2a. shows measured $S$-$T_{0}$ rotations in both charge sectors, and Fig. 2b.
shows the results of dephasing sine (Eq. 4) fits to each time-trace in the
data. We observe a clear distinction in $T_{2}^{*}$ based on detuning
(${\sim}6$ $\mu$s in (3,1,0), ${\sim}4.5$ $\mu$s in (3,0,1) and ${\sim}1$
$\mu$s at the charge transition), as well as a gradual decrease in $S$-$T_{0}$
rotation frequency (from ${\sim}2.41$ to ${\sim}2.12$ MHz) as detuning is
ramped from (3,1,0) to (3,0,1) that appears to flatten off farther from the
charge transition. We note that the $T_{2}^{*}$ values obtained using this fit
are slightly longer than the values reported for the data in Fig. 2c, and
conclude that this is expected of data averaged well below the ergodic limit
Madzik2020  (see discussion on Supplementary Fig. 3 for more information).
Overall, these observations demonstrate that the electron is well-isolated in
QD1 or QD2 because the qubit is shown to experience unique
$\ket{S}$-$\ket{T_{0}}$ rotation frequencies due to unique $\Delta g$
pertaining to each QD.

Figure 1: Singlet-triplet qubit in a triple quantum dot device. a. 3D profile
of the device showing the 28Si quantum well (light blue) buried within SiGe
(dark blue), overlapping metal gates (maroon) and oxide (gray) (above). 2D
overview of the device with relevant gates composing the SET and the shuttling
QDs encircled (below). b. Cartoon schematic of the electrostatic potential in
the shuttling QDs depicting the charge transition between (4,0,0) and (3,1,0).
The electron reservoir is shown under accumulation gate AC0 in purple.
Electron spins ordered on energy levels (light blue horizontal lines) are
moved between adjacent QDs by applying voltages to the plunger (P) and barrier
(B) gates in light blue. Voltages on other gates in dark blue are nominally
static. c. Energy diagram of the $S$-$T_{0}$ qubit in an applied magnetic
field. $J$ is defined between $\ket{S}$ and $\ket{T_{0}}$ states. $\Delta
E_{\mathrm{Z}}$ separates the $\ket{\uparrow\downarrow}$ and
$\ket{\downarrow\uparrow}$ states. $\ket{\uparrow\downarrow}$ is arbitrary in
permutation and cannot be distinguished from $\ket{\downarrow\uparrow}$, but
is used throughout for simplicity. d. $S$-$T_{0}$ Bloch sphere corresponding
to the energy diagram in c. showing orthogonal energy components $J$ and
$\Delta E_{\mathrm{Z}}$ e. Charge stability map formed by varying P0 and P1
gate voltages showing charge transitions between QD0 and QD1. The qubit is
reset in (3,0,0) for 140 $\mu$s (R), then loaded into (4,0,0) (L), rolled near
the (4,0,0)–(3,1,0) PSB region (approximate location outlined in pink) to
prepare for transfer (P), and then transferred by rapid adiabatic passage to
(3,1,0) (Q). The spin state is measured by latched PSB at the (4,0,0)-(4,1,0)
boundary (M). Figure 2: Shuttling and inhomogeneous dephasing. a. Cartoon
representation of an electron being shuttled from QD1 to QD2. b. Singlet
return probability ($\ket{S}$%) during shuttling the detuning space
surrounding the (3,1,0)-(3,0,1) charge transition. Three colored dots
represent P1-P2 detunings corresponding to measurements shown in c. c.
Inhomogeneous dephasing of the singlet return probability over time spent at
three detunings: (3,1,0) (gold), (3,0,1) (pink) and at the charge transition
(violet). Data at $B_{0}\approx 0$ (50) mT are shown as small translucent
(large opaque) circles, with decay envelope fits to Eq. 3 overlaid as dashed
(solid) lines. Curves are vertically offset for clarity. Full fit parameter
values and uncertainties are tabulated in Supplementary Table 2. Figure 3:
Repeated shuttling experiments. a. Pulse sequence diagram for repeated
shuttles. A $\ket{S}$ state is prepared, undergoes shuttling between QD1 and
QD2, and then its final spin state is measured. b. Cartoon representation of
the exchange interaction between QD0 and QD1,2 used to carry out a $\pi_{z}$
pulse for dynamical decoupling. c. Singlet return probability after $N$
shuttles (different colored markers) are executed over a range of time. A fit
of the form of Eq. 3 is overlaid (black curve). d. Singlet return probability
for an experiment similar to c., but with the dynamical decoupling sequence
shown in the inset applied. A fit of the form of Eq. 5 is overlaid (black
curve). e. Singlet return probability for a $\ket{\uparrow\downarrow}$ state
after $N$ shuttles executed over time. An exponential decay fit of the form of
Eq. 6 is overlaid (black curve). f., g., h. Repeated shuttles carried out at
fixed $t_{\mathrm{shuttle}}$ corresponding to the colored vertical dashed
lines in c., d., and e., respectively. $B_{0}\approx 0$ mT for the $\ket{S}$
shuttles (c.,d.,f.,g.) and $B_{0}=10$ mT for the $\ket{\uparrow\downarrow}$
shuttles (e., h.). All error bars shown represent the standard deviation of
the mean of the measurement.  Figure 4: Shuttling error estimates at large
$N$. Data is shown in teal markers where error bars represent standard
deviations of the mean of multiple repeated measurements. A phenomenological
fit of the form of Eq. 7 is overlaid (black curve). The range of
$T_{2}^{\mathrm{Hahn}}$ ($T_{1}$) based on analysis in the main text regarding
Fig. 3d. (Fig. 3e.) is shown in orange (red) shading, presented as a guide to
the eye. The top horizontal axis ($t_{\mathrm{shuttle}})$ is related to the
bottom ($N$ shuttles) by the expression
$t_{\mathrm{shuttle}}=N(2t_{\mathrm{wait}}+t_{\mathrm{ramp}})=14N\leavevmode\nobreak\
\mathrm{ns}$. Here, $B_{0}\approx 0$ mT. Supplementary Figure 1: Barrier-
enhanced coherent spin transfer. a.-e. Singlet return probability raster scans
of the charge space between (3,1,0) and (3,0,1) states for a range of barrier
pulse voltages dB2 $=0.1-0.2$ V used during the shuttle. The wait time in each
panel is 6 $\mu$s. f.-j. Singlet return over time as the detuning is ramped
from (3,1,0) to (3,0,1) along the trajectory defined by red arrow in the
corresponding raster plot shown in the panel directly to the left. $B_{0}=50$
mT and the detuning ramp time is 30 ns in all measurements shown.
Supplementary Figure 2: Shuttling from QD1 to QD2. a. Singlet return
probability over time as the detuning is ramped from (3,1,0) to (3,0,1). b.
Averaged fit parameters of $T_{2}^{*}$ dephasing time (left y-axis) and
$S$-$T_{0}$ rotation frequency (right y-axis) from a dephasing sine model (Eq.
4 with $\alpha=2$) fit to the data in a. Supplementary Figure 3: $T_{2}^{*}$
repeated measurements spanning 1 hour. a.-c. Repeated time traces of singlet
return probability for (3,1,0), (3,0,1) and the charge transition,
respectively, at $B_{0}=50$ mT. d.-f. Repeated time traces of singlet return
probability for (3,1,0), (3,0,1) and the charge transition, respectively, at
$B_{0}\approx 0$ mT. g. Gaussian $T_{2}^{*}$ fits to each line trace over time
in a.-c. averaged over an increasing number of line traces. The averaging time
on the horizontal axis is derived from the lab time accumulated over the
course of data acquisition, including dead time to read out and send
information to the computer. $T_{2}^{*}$ fits approach the ergodic limit after
about 10 minutes of averaging. Supplementary Figure 4: $S$-$T_{0}$
oscillations in an external magnetic field for (3,1,0) and (3,0,1) charge
configurations. a.-b. Singlet return probability as $B_{0}$ is ramped. The
qubit is held for a variable wait time in each of (3,1,0) and (3,0,1). c.-d.
Corresponding normalized FFTs to the data shown in a.,b. showing $S$-$T_{0}$
qubit rotation frequency.