Introduction

Spin qubits are a compelling candidate for the foundation of a quantum register network, having demonstrated single and two-qubit gate fidelities over 99%1,2,3,4. By reducing the spinful 29Si nuclear background in silicon, isotopic enrichment enables coherent spin qubits to be hosted in foundry-compatible gate-defined quantum dots (QDs)5,6,7, streamlining integration with existing and rapidly advancing industrial silicon technology. Since spin qubit operations are carried out using short-range interactions often involving nearest neighbors8,9, 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 processor-length scales by carrying out qubit operations at dedicated local nodes linked by interconnected hardware10,11.

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 transducers12,13,14. Another approach is to physically move the spin qubit, for example, by using surface acoustic waves to form a moving QD15,16, or by shuttling the spin controllably through an array of gate-defined electrostatic potential traps17,18,19,20. 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 error sources during shuttling is therefore required. For a spin qubit device composed of millions of qubits, for example, arranged on a crossbar array21,22 or a sparse qubit node array11,23, thousands to tens of thousands of shuttling operations may need to be performed to link sites on either end of a macroscopic processor. At present, error dynamics are not well quantified out to the large number of shuttles needed to establish these long-range interactions.

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 < 103 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 additional shuttle errors are incoherent but do not contribute appreciably to losses until N > 103, meaning that errors due to shuttling are small.

Previous works exploring coherent shuttling span a variety of different systems ranging from enriched Si MOS24, natural Si/SiGe25, enriched Si/SiGe20, Ge/SiGe18 and AlGaAs/GaAs17,26 heterostructures; most were performed in external magnetic fields of about 0.2 ~ 1 T and a few included micromagnets. Here, we investigate shuttling errors in the low applied external magnetic field regime (~mT) without a micromagnet, where we expect spin-orbit-induced field gradients between QD locations to be reduced. Additionally, we operate with relatively large tunnel coupling between QDs to maintain a low probability of nonadiabatic charge transfer during shuttles. Understanding the various noise mechanisms encountered during shuttling in this regime can aid the development of targeted mitigation strategies that will enable the completion of a large number of shuttles, allowing an opportunity for linking localized27,28 qubit nodes on a sparse qubit array.

Figure 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 nQDi at each QDi are primarily controlled by voltages on plunger gates Pi, where i = 0, 1, 2 is the QD index. We denote the total charge occupation across the triple QD array by (nQD0, nQD1, nQD2). 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).

Fig. 1: Singlet-triplet qubit in a triple quantum dot device.
figure 1

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-T0 qubit in an applied magnetic field. J is defined between \(\left\vert S\right\rangle\) and \(\left\vert {T}_{0}\right\rangle\) states. ΔEZ separates the \(\left\vert \!\uparrow \downarrow\! \right\rangle\) and \(\left\vert\! \downarrow \uparrow \!\right\rangle\) states. \(\left\vert \!\uparrow \downarrow \!\right\rangle\) is arbitrary in permutation and cannot be distinguished from \(\left\vert\! \downarrow \uparrow \!\right\rangle\), but is used throughout for simplicity. (d). S-T0 Bloch sphere corresponding to the energy diagram in (c). showing orthogonal energy components J and ΔEZ (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 μs (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).

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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. Figure 1c shows an energy diagram of the remaining two-spin system illustrating the relationship between the singlet (\(\left\vert S\right\rangle\)) and triplet (\(\left\vert {T}_{0}\right\rangle ,\,\left\vert {T}_{+}\right\rangle ,\,\left\vert {T}_{-}\right\rangle\)) states in a small magnetic field. We employ a singlet-triplet (S-T0) qubit29, where the qubit is encoded in the m = 0 subspace of the 4-level system defined by linear combinations of the \(\left\vert\! \uparrow \downarrow\! \right\rangle\) and \(\left\vert \!\downarrow \uparrow\! \right\rangle\) states. Its logical states are defined as the eigenstates of the system in the limit of large exchange:

$$\begin{array}{lll}\left\vert 0\right\rangle =\left\vert S\right\rangle \,=\,\frac{1}{\sqrt{2}}(\left\vert\! \uparrow \downarrow \!\right\rangle -\left\vert \!\downarrow \!\uparrow \right\rangle )\\ \left\vert 1\right\rangle =\left\vert {T}_{0}\right\rangle \,=\,\frac{1}{\sqrt{2}}(\left\vert \!\uparrow \!\downarrow \right\rangle +\left\vert \!\downarrow \!\uparrow \right\rangle )\end{array}$$
(1)

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 of spin-to-charge conversion through Pauli spin blockade to measure the qubit spin state30. 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, ΔEZ (Fig. 1d). We use voltage pulses applied to the B1 barrier gate and adjacent P0 and P1 QD gates at the exchange symmetric operating point31,32 to achieve rotations between \(\left\vert \!\uparrow \downarrow \!\right\rangle\) and \(\left\vert \!\downarrow \uparrow\! \right\rangle\). When the electrons are well isolated and exchange is turned off, \(\left\vert S\right\rangle\) to \(\left\vert {T}_{0}\right\rangle\) rotations are driven by33

$$\Delta {E}_{{\rm{Z}}}=\Delta g{\mu }_{{\rm{B}}}B,$$
(2)

where g is the g-factor, μB is the Bohr magneton, and B is the magnetic field. Differences in local magnetic fields or local g-factors both contribute to ΔEZ. 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 field34. 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 interactions33.

Results

Coherent spin transfer

Two established approaches for spin shuttling are conveyor mode25,35,36 and bucket brigade mode3,17,18,26,37,38. 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 \(\left\vert S\right\rangle\) state in QD0 and QD1 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 VP1 and VP2 (ramping in opposite directions), while simultaneously pulsing VB2 to lower the potential barrier and facilitate adiabatic transfer (Fig. 2a) (see discussion on Supplementary Fig. 1 for more details). In principle, this shuttling method is agnostic to spin qubit encoding, provided individual spin control is possible. The tunnel coupling between QD1 and QD2 is estimated ~52 GHz with negligible probability of completing a nonadiabatic transition (see discussion on Supplementary Fig. 2b for more details). We hold VB1 fixed to maintain negligible residual J with the electron in QD0 and ensure rotations about the Bloch sphere are driven by the dominant ΔEZ interaction (see Supplementary Table I for a voltage level table).

Fig. 2: Shuttling and inhomogeneous dephasing.
figure 2

a Cartoon representation of an electron being shuttled from QD1 to QD2. b Singlet return probability (\(\left\vert S\right\rangle\)%) 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 B0 ≈ 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 II.

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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 B0 = 50 mT in the plane of the Si quantum well to intentionally drive strong S-T0 rotations in both charge regions, showing the spin is coherently transferred. Figure 2b shows singlet return probability following a 6 μs 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 within each QD modifying ΔEZ as QD1 and QD2 are shaped by applied voltages, and the local interface disorder sampled by the electron varies33. The charge transition between (3,1,0) and (3,0,1) appears as a diagonal strip of constant ~50% singlet return probability 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.

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 \(\left\vert S\right\rangle\) state and letting it evolve under ΔEZ. This measurement is sensitive to phase-flip errors assuming spin-flip errors are much smaller. We measure the singlet return probability over time, averaged over 1 hour total to ensure the ergodic limit was reached39 (Supplementary Fig. 3). Figure 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 (B0 ≈ 0 mT, with a remanent field of up to a few mT due to trapped flux in the superconducting magnet) such that dephasing is expected to be dominated by fluctuations in the local Overhauser field29, where no S-T0 rotations are observed. We fit the data to a decay profile of the form

$${f}_{{T}_{2}^{* }}(t)=A{e}^{-{\left(t/{T}_{2}^{* }\right)}^{\alpha }}+{y}_{0},$$
(3)

where A is the amplitude, y0 is the vertical offset, \({T}_{2}^{* }\) is the inhomogeneous dephasing time, and α is the decay exponent. The resulting dephasing times and fit uncertainties are \({T}_{2}^{* }(3,1,0)=4.60\pm 0.02\,\mu {\rm{s}}\), \({T}_{2}^{* }(3,0,1)=3.86\pm 0.02\,\mu {\rm{s}}\), and \({T}_{2}^{* }({\rm{transition}})=5.12\pm 0.03\,\mu {\rm{s}}\), and are consistent with previously measured quantities for QDs in 800 ppm 29Si6,40,41. The variation among \({T}_{2}^{* }{\rm{s}}\) suggests local sources of dephasing affect the spin in each QD, including the wavefunction overlap with nearby 29Si and 73Ge. We interpret the longer \({T}_{2}^{* }({\rm{transition}})\) 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 QD242. This increase in \({T}_{2}^{* }\) when the second electron interacts with a larger number of nuclear spins when hybridized is consistent with the central limit theorem, which scales as \({T}_{2}^{* } \sim \sqrt{{N}_{{\rm{spins}}}}\), where Nspins is the number of spinful nuclei lying within the electron wavefunction43. The resulting decay exponent fits, α(3, 1, 0) = 1.92 ± 0.02, α(3, 0, 1) = 2.16 ± 0.02, and α(transition) = 2.12 ± 0.02, resemble Gaussian profiles as expected of inhomogeneous nuclear spin dephasing29.

Next, we apply a small external field of B0 = 50 mT to amplify and study the effect of charge noise acting on ΔEZ. Clear S-T0 oscillations are observed and fit to a decaying sine function of the form

$${f}_{{T}_{2}^{* }}(t)=A{e}^{-{\left(t/{T}_{2}^{* }\right)}^{\alpha }}\sin \left(2\pi ft\right)+{y}_{0},$$
(4)

where f is the S-T0 rotation frequency. The resulting dephasing times are \({T}_{2}^{* }(3,1,0)=3.82\pm 0.12\,\mu {\rm{s}}\), \({T}_{2}^{* }(3,0,1)=4.09\pm 0.03\,\mu {\rm{s}}\), and \({T}_{2}^{* }({\rm{transition}})=1.08\pm 0.01\,\mu {\rm{s}}\). We attribute the fast decay observed at the charge transition to the hybridization of the electron wavefunction split between QD1 and QD218,24, which enhances the qubit sensitivity to charge noise. Here, charge noise rapidly alters the extent of the wavefunction overlap within QD1 and QD2, each with its own distinct spin-orbit interactions. This manifests as increased fluctuations in ΔEZ experienced by qubit, which increases with magnetic field, driving faster S-T0 dephasing. This sensitivity to charge noise may be minimized by tuning the external magnetic field angle44, or by simply reducing the magnitude. For the remainder of this work concerning shuttling, we maintain a small or negligible field to reduce the susceptibility of the qubit to noise through ΔEZ.

Additionally, we observe changes to α that imply changes to the noise spectrum. The resulting decay exponent fits are α(3, 1, 0) = 0.90 ± 0.06, α(3, 0, 1) = 2.12 ± 0.06, and α(transition) = 1.14 ± 0.02, ranging from exponential to Gaussian forms. Since the noise spectrum and mechanisms dictate the observed decay profile45, the various α suggest 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, where the observed charge noise is often attributed to two-level systems (TLSs) that are distributed nonuniformly within the gate dielectric46,47. This is consistent with our assessment that charge noise dominates the fast \({T}_{2}^{* }({\rm{transition}})\), and suggests an increased susceptibility to charge noise in QD1 compared to QD2, since α(3, 1, 0) is exponential and α(3, 0, 1) is Gaussian. Overall, we observe qubit inhomogeneous dephasing to be primarily driven by fluctuations in the Overhauser fields generated by residual nuclear spins near 0 mT, and a combination of nuclear spins and charge noise at 50 mT.

Repeated spin shuttles

To understand the role of shuttle-induced errors, independent of intrinsic noise, we perform repeated spin shuttling operations to investigate errors due to 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 (2twait), and the ramp time (tramp) 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 tshuttle = N(2twait + tramp) (Fig. 3a).

Fig. 3: Repeated shuttling experiments.
figure 3

a Pulse sequence diagram for repeated shuttles. A \(\left\vert S\right\rangle\) 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 πz 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 \(\left\vert\! \uparrow \downarrow\! \right\rangle\) state after N shuttles executed over time. An exponential decay fit of the form of Eq. (6) is overlaid (black curve). f–h Repeated shuttles carried out at fixed tshuttle corresponding to the colored vertical dashed lines in (c–e), respectively. B0 ≈ 0 mT for the \(\left\vert S\right\rangle\) shuttles (c, d, f, g) and B0 = 10 mT for the \(\left\vert\! \uparrow \downarrow \!\right\rangle\) shuttles (e, h). All error bars shown represent the standard deviation of the mean of the measurement.

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To investigate dephasing during shuttling, we vary twait for a series of fixed N carried out within tshuttle (Fig. 3c), with B0 ≈ 0 mT to minimize the charge noise-induced field-dependent dynamics on ΔEZ 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 \({T}_{2}^{* }=5.07\pm 0.07\,\mu {\rm{s}}\), which is consistent with \({T}_{2}^{* }({\rm{transition}})\) measured in the absence of shuttling at B0 = 0 mT (Fig. 2c). This indicates the spin samples a similar noise environment when shuttled and hybridized. This observation suggests the error introduced by the act of shuttling, or shuttle error, is low. The decay profile is nearly Gaussian with α = 1.82 ± 0.07, which points to the dominant dephasing mechanism being nuclear spin fluctuations.

Next, we employ a standard spin echo technique29 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 inspect the role of shuttle errors on observed decay. For this experiment, B0 ≈ 0 mT to minimize field-dependent effects on ΔEZ. We prepare the \(\left\vert S\right\rangle\) 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 πz 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 1/2tshuttle. Figure 3d shows the singlet return probability after implementing the spin echo sequence during shuttling for a range of N completed within tshuttle. We observe a single decay profile independent of N and fit the data to a generic decay function of the form

$${f}_{{T}_{2}^{{\rm{Hahn}}}}(t)=A{e}^{-{\left(t/{T}_{2}^{{\rm{Hahn}}}\right)}^{\alpha }}+{y}_{0},$$
(5)

similar to Eq. (3). The fit yields \({T}_{2}^{{\rm{Hahn}}}=84.48\pm 7.82\,\mu {\rm{s}}\) which is reasonably consistent with previously measured values in Si/SiGe heterostructures48,49, and indicates sources of noise contributing to dephasing have been reduced. The fit converges with an exponent of α = 0.64 ± 0.06, which deviates from α commonly observed in the range of 1 ≲ α ≲ 229,48,50, suggesting a different noise spectrum is introduced. Sub-exponential T2 decays have been previously observed51 and simulated52 in other semiconductor-based spin systems, such as NV− centers in diamond, with origins tracing back to the particular spin dynamics of the bath. In our system, the exact mechanisms driving decay are as of yet unknown but may be related to bath dynamics at low magnetic field.

Next, we investigate the role of spin relaxation during shuttling since spin-flip mechanisms are known error sources in Si QDs53. We prepare an eigenstate \({\left\vert \uparrow \downarrow \right\rangle }_{(3,1,0)}\) of the ΔEZ interaction and monitor return of this state while repeatedly shuttling between QD1 and QD2. This measurement is sensitive to spin-flip errors but not phase-flip errors. We use a small applied field of B0 = 10 mT to avoid mixing with \(\left\vert {T}_{-}\right\rangle\) during the (3,1,0) charge state preparation. Figure 3e shows singlet return probability of the shuttled \(\left\vert \uparrow \downarrow \right\rangle\) state for a range of N completed within tshuttle. We observe a single decay profile independent of N and fit the data to an exponential decay function of the form

$${f}_{{T}_{1}}(t)=A{e}^{-t/{T}_{1}}+{y}_{0}$$
(6)

with a range of assumptions placed on y0, the singlet return saturation, since it is not explicitly measured. The fit results in T1 = 6.22 ± 0.71 ms, where the uncertainty represents fit variation based on y0, and lies within the broad range of T1 measured for electrons confined to Si QDs, typically falling between milliseconds and seconds41,54,55. A variety of mechanisms generally stemming from spin-orbit and spin-valley interactions contribute to spin relaxation in Si QDs56. Due to the natural disorder and inversion asymmetry at the Si/SiGe interface, the small in-plane B0 gives rise to spin-orbit effects33 that can induce spin-flip relaxation over time. At B0 = 0~10 mT, we expect a small but nonzero influence from the spin-valley hotspot measured near 190 mT in QD0, corresponding to a valley-splitting energy of ~21 μeV in QD0 (see discussion on Supplementary Fig. 4), which could drive a faster relaxation rate (\({T}_{1}^{-1}\))53. Spin-valley hotspots for the other QDs are higher in energy and would likely play a smaller role. Few references for T1 in the low-field limit exist; however, the observation that \({T}_{1} > {T}_{2}^{{\rm{Hahn}}}\) in our device shows that incoherent errors due to spin-flip processes are relatively small compared to coherent errors arising from magnetic noise during shuttling for N ≲ 103.

From the data presented in Fig. 3c–e, we anticipate the shuttle 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 tshuttle while varying N such that the effect of time-dependent dephasing and errors is constant with N. Figure 3f–h show the measured singlet return probability at three fixed tshuttle corresponding to the shuttling experiments shown in the insets of Fig. 3c–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 ≈ 103 (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 10−4, an improvement from recently reported shuttling errors.

Due to the low shuttling error inferred from time-independent measurements, 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 technique out to the maximum N possible within our instrument bandwidth capabilities. We compose a shuttling experiment at B0 ≈ 0 mT, with the minimum possible 2twait = 4 ns and increase the repeated shuttles out to N = 4 × 105, spanning a total time of tshuttle = 5.6 ms, where tshuttle(N) = 14N ns. The singlet return probability over N, and equivalent tshuttle, is shown in Fig. 4. We consider the observed decay in the context of estimated \({T}_{2}^{{\rm{Hahn}}}\) and T1 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 < 103, the \({T}_{2}^{{\rm{Hahn}}}\) decay profile agrees with the data and implies the decay is dominated by the nontrivial magnetic spectral noise acting on ΔEZ. For a large number of shuttles N > 103, 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 losses are due to an incoherent spin-flip relaxation process. However, the process causes decay on a timescale faster than the measured T1 (for N < 103), indicating a small, but distinct, incoherent spin-flip type shuttle error that is only observable when shuttling a large number of times (N > 103).

Fig. 4: Shuttling error estimates at large N.
figure 4

Data are 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 \({T}_{2}^{{\rm{Hahn}}}\) (T1) 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 (tshuttle) is related to the bottom (N shuttles) by the expression tshuttle = N(2twait + tramp) = 14 Nns. Here, B0 ≈ 0 mT.

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

$${f}_{{N}^{* }}(N)=A{e}^{-{(N/{N}^{* })}^{\alpha }}+{y}_{0},$$
(7)

which defines the characteristic decay in units of N (N*). We find N* = (1.92 ± 0.32) × 104 shuttles can be completed within a shuttle error-limited characteristic decay period, 1–2 orders of magnitude more than recently demonstrated with shuttled spins18,20. The observation that the function fits poorly for smaller N < 103 suggests the influence of shuttle errors do not become apparent until after the completion of N ≈ 103 shuttles. The fit results in α = 0.41 ± 0.03 and, 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.

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 Si3,24 and in Ge18 QDs. Using an 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 and to provide an independent estimate of the shuttle error. Independent of shuttling, we identify magnetic noise from Overhauser field fluctuations as the dominant source of dephasing in the absence of external fields (B0 ≈ 0 mT) and a combination of nuclear bath noise and charge noise at finite field (B0 = 50 mT). When the spin is repeatedly shuttled, we observe little change in the decay at B0 ≈ 0 mT, indicating that shuttling operations marginally contribute to losses. Using a standard spin echo technique, we show the coherence time during shuttling may be extended owing to the reduction of magnetic noise on ΔEZ and is limited by a different noise spectrum related to hyperfine interactions indicated by a subexponential \({T}_{2}^{{\rm{Hahn}}}\) decay profile. Residual uncorrelated magnetic noise on \({T}_{2}^{{\rm{Hahn}}}\) timescales is determined to be the limiting time-dependent contribution since \({T}_{2}^{{\rm{Hahn}}} < {T}_{1}\). Additionally, we find that \({T}_{2}^{* }\), \({T}_{2}^{{\rm{Hahn}}}\), and T1 appear to be N-independent in the small N < 103 regime, where shuttle errors do not contribute appreciably. Fitting the dephasing and relaxation data in this regime, (Fig. 3–h), we make a rudimentary yet encouraging shuttle error estimate of 10−4.

By executing shuttle operations in the large N regime with a maximum of N = 4 × 105 shuttles, we identify incoherent spin-flip type shuttle errors which become apparent after many shuttle operations. In the small N (<103) regime, shuttle errors are negligible, and decay is primarily limited by magnetic dephasing on \({T}_{2}^{{\rm{Hahn}}}\) timescales. In the large N (>103) regime, a shuttle-induced, or N-dependent mechanism driving spin relaxation becomes apparent, causing decay on a timescale faster than the measured T1 (for N < 103). The exact mechanisms driving shuttle errors are unclear given the evidence, but their origins can be narrowed down. Nonadiabatic charge transfer is expected to contribute imperceptibly small errors due to the large tunnel coupling between QD1 and QD2, as previously noted. We also expect errors arising from electron-nuclear spin flip-flops when repeatedly ionizing QD1 and QD2 to be below the measurable shuttle error bounds (<10−4)28.

On the other hand, the physical motion of the electron through the transition may give rise to additional relaxation channels. Charge noise is known to induce spin relaxation through driven transverse spin-orbit fluctuations, which would be most pronounced at the QD charge transition55. Furthermore, the strong charge hybridization at the transition could lead to an effective reduction in valley or orbital splitting as well as changes in the spin-valley coupling, which could enhance spin relaxation though spin-valley hotspot-like mechanisms53. Reduced valley splitting or rapid change of valley phase when shuttling between QDs may also increase susceptibility to excitations outside of the valley ground state during transfer, but recent theoretical work suggests encouraging evidence that optimized control schedules may reduce this likelihood during spin shuttling57. In the above scenarios, the probability of these spin-flip errors occurring would increase as more shuttles are appended, or as the shuttled electron spends more time near the charge transition. While these values have not been measured explicitly in the low magnetic field regime, they can be targeted as the focus of future studies expanding upon spin shuttling errors.

Looking ahead, error estimates based on the \({T}_{2}^{* }\) and T1 decays are the more practical expectation for building up a long-range spin shuttling platform. Focusing on the timescales quantified in this work (\({T}_{2}^{* }\) and T1), we make estimates of the error per shuttle \(p=1-{e}^{-{(14{\rm{ns}}/\tau )}^{\alpha }}\) based on the assumption of the limiting decay timescale τ during a 14 ns shuttling operation, summarized in Table 1. If all other noise sources were remediated and the fluctuating field gradient is the only remaining noise mechanism so that decay is limited by \({T}_{2}^{* }\), we extrapolate the shuttling error to be p = 2.2 × 10−5. If noise on ΔEZ could also be reduced to the level such that spin-flip relaxation on the T1 timescale limits decay, the shuttle error is estimated even lower at p = 2.3 × 10−6. These results highlight the potential for high-fidelity shuttling as a promising pathway to remote spin entanglement in spin-based quantum computing platforms.

Table 1 Shuttle errors based on decay fit calculated from the time it takes to complete one shuttle (\(p=1-{e}^{-{(14{\rm{ns}}/\tau )}^{\alpha }}\))
Full size table

Methods

Device details

The device was fabricated by Intel Corporation and is based on a Si/Si0.7Ge0.3 heterostructure6, 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 ~120 nm, the Si well thickness is ~5 nm, and the SiGe barrier thickness is ~50 nm.

Experimental setup

We use a closed-loop dry cryostat Bluefors LD400 dilution refrigerator with a base temperature of 7 mK. The electron temperature is approximately 250 mK. We use a heterojunction bipolar transistor operating at −1.056 V emitter bias on the readout SET line to perform cryogenic preamplification58. DC voltage to the gates is supplied by a QDevil QDAC II, and fast AC pulses are 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 use 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.