Abstract
Spinbased quantum processors in silicon quantum dots offer highfidelity single and twoqubit operation. Recently multiqubit devices have been realized; however, manyqubit demonstrations remain elusive, partly due to the limited qubittoqubit connectivity. These problems can be overcome by using SWAP gates, which are challenging to implement in devices having large magnetic field gradients. Here we use a primitive SWAP gate to transfer spin eigenstates in 100 ns with a fidelity of \({\bar{F}}_{{\rm{SWAP}}}^{{\rm{(p)}}}=98 \%\). By swapping eigenstates we are able to demonstrate a technique for reading out and initializing the state of a double quantum dot without shuttling charges through the quantum dot. We then show that the SWAP gate can transfer arbitrary twoqubit product states in 300 ns with a fidelity of \({\bar{F}}_{{\rm{SWAP}}}^{{\rm{(c)}}}=84 \%\). This work sets the stage for manyqubit experiments in silicon quantum dots.
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Introduction
Solid state quantum processors based on spins in silicon quantum dots are emerging as a powerful platform for quantum information processing.^{1,2} Highfidelity single and twoqubit gates have recently been demonstrated^{2,3,4,5,6} and large extendable qubit arrays are now routinely fabricated.^{7} However, twoqubit gates are mediated through nearestneighbor exchange interactions,^{1,8} which require direct wavefunction overlap. This limits the overall connectivity of these devices and is a major hurdle to realizing error correction,^{9} quantum random access memory,^{10} and multiqubit quantum algorithms.^{11} To extend the connectivity, qubits can be shuttled around a device using quantum SWAP gates, but phase coherent SWAPs have not yet been realized in silicon quantum dot devices.^{2,3,4,5,6}
Here, we demonstrate a singlestep resonant SWAP gate. We first use the gate to efficiently initialize and readout our double quantum dot. We then show that the gate can move spin eigenstates in 100 ns with average fidelity \({\bar{F}}_{{\rm{SWAP}}}^{{\rm{(p)}}}=98 \%\). Finally, the transfer of arbitrary twoqubit product states is benchmarked using state tomography and Clifford randomized benchmarking,^{5,6,12} yielding an average fidelity of \({\bar{F}}_{{\rm{SWAP}}}^{{\rm{(c)}}}=84 \%\) for gate operation times of ~300 ns. Through coherent spin transport, our resonant SWAP gate enables the coupling of nonadjacent qubits, thus paving the way to large scale experiments using silicon spin qubits.
Results
Device architecture
In this work, we use two sites of a quadruple quantum dot fabricated on a ^{28}Si/SiGe heterostructure (inset of Fig. 1a).^{13} Electric dipole spin resonance^{14,15} enables singlespin control and an onchip micromagnet detunes the frequency of each spin to enable siteselective control.^{13,16} For demonstration purposes, we use two dots in the device with qubits accumulated under plunger gates \({P}_{3}\) and \({P}_{4}\). We hereafter refer to the two qubits as \({Q}_{3}\) and \({Q}_{4}\), respectively. This naming is consistent with our previous work where operation of the device was first demonstrated.^{13} The charge stability diagram of this DQD is shown in Fig. 1a and quantum control is performed in the \(({N}_{3},{N}_{4})=(1,1)\) charge configuration, where \({N}_{i}\) denotes the number of electrons on dot \(i\). We measure the state of \({Q}_{4}\) through spinselective tunneling to a drain reservoir accumulated beneath gate \({D}_{3}\).^{17} State initialization is also performed through spinselective tunneling and the loading fidelity is limited to 95% by the 110 mK electron temperature.
Twoqubit Interactions
There are two modes of operation for the resonant SWAP gate demonstrated in this article. First, a projectionSWAP can be used to transfer spin eigenstates between quantum dots. The projectionSWAP enables rapid initialization and readout of inner sites in an array that are not directly connected to the leads—meaning they cannot be directly initialized or measured. Secondly, a coherentSWAP can be used to transfer arbitrary quantum states between quantum dots, thus allowing the rearrangement of qubits in the array. A coherentSWAP is important for performing multiqubit algorithms or error correction in devices with limited qubittoqubit connectivity.^{18} The coherent and projectionSWAP gates are realized using the same resonant SWAP gate; however, the coherentSWAP requires more stringent calibration.
In our device architecture, twoqubit gates are mediated through the exchange interaction \({J}_{i,i+1}({V}_{{\rm{B}}i+1})\), which is proportional to the wavefunction overlap between the two adjacent qubits \(i\) and \(i+1\). This wavefunction overlap, and thus the exchange interaction, is controlled by adjusting the barrier gate voltage \({V}_{Bi+1}\). Here, we realize our SWAP gate through a resonant drive on that barrier at a frequency \({f}_{{\rm{SWAP}}}\) according to the formula \({V}_{{{B}}i+1}(t)={V}_{{{B}}i+1}^{({\rm{dc}})}+{V}_{{{B}}i+1}^{({\rm{ac}})}\cos (2\pi {f}_{{\rm{SWAP}}}t+\phi )\) where \({V}_{{{B}}i+1}^{({\rm{ac}})}\) is the amplitude of the drive and \({V}_{{{B}}i+1}^{({\rm{dc}})}\) is a static offset voltage. We note that the exchange interaction and the gate can also be modulated by varying the double quantum dot detuning.^{8}
In the absence of a magnetic field gradient, when \({J}_{i,i+1}\gg {\gamma }_{\mathrm e} {B}_{i}^{{\rm{tot}}}{B}_{i+1}^{{\rm{tot}}}\), where \({\gamma }_{\mathrm e}\) is the electronic gyromagnetic ratio and \({B}_{i}^{{\rm{tot}}}\) is the magnetic field at dot \(i\), the two qubits directly undergo SWAP oscillations.^{8,19,20,21} However, many spin qubit devices rely on large magnetic field gradients^{2,4,6,13} and our device has \({\gamma }_{\mathrm e}{B}_{3}^{{\rm{tot}}}\) = 16.949 GHz and \({\gamma }_{\mathrm e}{B}_{4}^{{\rm{tot}}}\) = 17.089 GHz at an external magnetic field of 410 mT. In this regime, the exchange interaction leads to a CPHASElike evolution.^{2,22,23} To recover the twoqubit SWAP oscillations in the presence of such large magnetic field gradients, we effectively rotate out the gradient by applying an exchange pulse that is resonant with the difference frequency of the two qubits (\(2\pi {f}_{{\rm{SWAP}}}={\gamma }_{\mathrm e} {B}_{i}^{{\rm{tot}}}{B}_{i+1}^{{\rm{tot}}}\)).^{24,25} This can be qualitatively understood as stroboscopically applying exchange whenever the evolution due to the magnetic field gradient returns to its initial state.
Resonant modulation of \({V}_{{{B}}i+1}\) at the difference frequency of the two qubits will drive Rabi rotations in the \(\left{\phi }_{3},{\phi }_{4}\right\rangle \in \{\left\uparrow \downarrow \right\rangle ,\left\downarrow \uparrow \right\rangle \}\) subspace, while leaving the fully spinpolarized states unaffected. A \(\pi\)pulse in this subspace is a SWAP gate up to additional phases on the state of each qubit. These phases do not affect the operation of the projectionSWAP, but will affect the coherentSWAP. The theory describing these phase shifts and a procedure for calibrating them out is outlined in Supplementary Discussions I–III (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory). Our resonant SWAP gate is therefore efficiently realized through a single RF burst on the barrier gate between qubits \(i\) and \(i+1\).
ProjectionSWAP
We first describe how the projectionSWAP gate can be used for readout of an interior site in our device using the protocol outlined in Fig. 1b. In a typical measurement cycle, after quantum control is performed at (1,1), \({Q}_{4}\) is read out through spinselective tunneling to the leads [blue triangle in Fig. 1a]. If \({Q}_{4}\) is in the \(\left\uparrow \right\rangle\) state, the electron has enough energy to tunnel off of the dot and is replaced by a lowerenergy electron in the \(\left\downarrow \right\rangle\) state. However, if \({Q}_{4}\) is in the \(\left\downarrow \right\rangle\) state, no tunneling occurs.^{17} Any charge hops are detected by monitoring the current through an adjacent charge sensor [\({I}_{{\rm{S}}2}\) in Fig. 1a]. If a charge hopping event is detected, we record the spin state as \(\left\uparrow \right\rangle\). Regardless of its initial state, \({Q}_{4}\) is left in the \(\left\downarrow \right\rangle\) state after measurement to within initialization errors. We next tune the device back into the (1,1) charge configuration and apply the resonant SWAP gate. In this process, \({Q}_{3}\)’s state is mapped on to \({Q}_{4}\) and \({Q}_{3}\) is left in the \(\left\downarrow \right\rangle\) state. We once again measure \({Q}_{4}\) to infer \({Q}_{3}\)’s original state. This measurement protocol leaves the DQD in the \(\left\downarrow \downarrow \right\rangle\) state.
To calibrate the SWAP gate, the system is initialized in the \(\left{\phi }_{3},{\phi }_{4}\right\rangle =\left\downarrow \downarrow \right\rangle\) state through spindependent tunneling from a Fermi reservoir into Q_{4} (ref. ^{4}) is then flipped using an \(X\) gate. The SWAP is implemented by driving gate \({B}_{4}\) with an RF burst at frequency \({f}_{{\rm{SWAP}}}\) and duration 600 ns. Figure 2a shows the spinup probability of \({Q}_{4}\), \({P}_{4,\uparrow }\) as a function of \({f}_{{\rm{SWAP}}}\) and \({V}_{{\rm{B}}4}^{({\rm{ac}})}\). For small \({V}_{{\rm{B}}4}^{({\rm{ac}})}\), there are no measureable SWAP oscillations. At around \({V}_{B4}^{({\rm{ac}})}\) = 10 mV, coherentSWAP oscillations in \({P}_{4,\uparrow }\) appear. The pattern is symmetric about \({f}_{{\rm{SWAP}}}\) = 140 MHz. For a 600 ns burst at 140 MHz, a SWAP is achieved with \({V}_{B4}^{{\rm{ac}}}\) = 10 mV. To minimize the SWAP time, we fix \({f}_{{\rm{SWAP}}}\) = 140 MHz and vary \({V}_{{\rm{B}}4}^{({\rm{ac}})}\) and the drive time (\({t}_{{\rm{SWAP}}}\)) in Fig. 2b. Each alternating bright fringe corresponds to an even number of SWAPs. The minimum SWAP time shown here is 23 ns and is limited by the dynamic range of our control electronics. The coherence times \({T}_{2}^{* }\) are approximately 10 μs for both dots,^{13} which is long relative to these gate operation times.
We now demonstrate simultaneous quantum control, initialization, and readout of both dots using spintocharge conversion of only \({Q}_{4}\). Starting in the \(\left\downarrow \downarrow \right\rangle\) state, we apply a microwave burst with duration \({\tau }_{{\rm{p}}}\) and frequency \(f\). We measure \({Q}_{4}\) through spinselective tunneling to \({D}_{3}\), leaving \({Q}_{4}\) in the \(\left\downarrow \right\rangle\) state. We then apply a projectionSWAP to the qubits, mapping the state of \({Q}_{3}\) onto \({Q}_{4}\), and leaving \({Q}_{3}\) in the \(\left\downarrow \right\rangle\) state. \({Q}_{4}\) is then measured so that we can infer the state of \({Q}_{3}\). Once \({Q}_{4}\) is measured, the qubits are left in the \(\left\downarrow \right\rangle\) state, and the DQD is prepared for the next experiment. The spinup probability for both qubits is plotted as a function of \({\tau }_{{\rm{p}}}\) and \(f\) in Fig. 2c. The fringes observed are Rabi oscillations, whose spacing is largest when the qubits are on resonance. These data reveal a qubit difference frequency of 140 MHz, which is consistent with the twoqubit spectroscopy in Fig. 2a. These data show that we can initialize, control, and readout our DQD even though readout only occurs on \({Q}_{4}\).
To quantitatively study the projectionSWAP gate for readout purposes, we designed an experiment to be insensitive to state preparation and measurement (SPAM) errors. We prepare the qubits in one of the four spin eigenstates \({\left{\phi }_{3},{\phi }_{4}\right\rangle }_{{\rm{in}}}=\left\downarrow \downarrow \right\rangle\), \(\left\downarrow \uparrow \right\rangle\), \(\left\uparrow \downarrow \right\rangle\), and \(\left\uparrow \uparrow \right\rangle\) before applying the SWAP gate to the qubits \(N\) times as shown in Fig. 3. We expect the spinpolarized states to decay towards a mixed state with \({P}_{3,\uparrow }\) and \({P}_{4,\uparrow }\) = 0.5 for large \(N\). The antiparallel spin input states should flipflop between the \(\left\downarrow \uparrow \right\rangle\) and \(\left\uparrow \downarrow \right\rangle\) states for each additional SWAP we apply. \({P}_{3,\uparrow }\) and \({P}_{4,\uparrow }\) will then converge to 0.5 in the large \(N\) limit. The decay envelope is given by \({F}_{{s}_{3}{s}_{4}}^{({\rm{p}})N}\), where \({F}_{{s}_{3}{s}_{4}}^{({\rm{p}})}\) is the fidelity of the projectionSWAP on spin states \({s}_{3}\) and \({s}_{4}\) between \({Q}_{3}\) and \({Q}_{4}\). Fitting these curves we find an average fidelity of \({F}_{\downarrow \uparrow }^{({\rm{p}})}={F}_{\uparrow \downarrow }^{({\rm{p}})}=96.5 \%\) for \({\left{\phi }_{3},{\phi }_{4}\right\rangle }_{{\rm{in}}}=\left\downarrow \uparrow \right\rangle\) or \(\left\uparrow \downarrow \right\rangle\). In cases where both qubits have the same initial state \({\left{\phi }_{3},{\phi }_{4}\right\rangle }_{{\rm{in}}}=\left\downarrow \downarrow \right\rangle\) or \(\left\uparrow \uparrow \right\rangle\), we achieve fidelities of \({F}_{\downarrow \downarrow }^{({\rm{p}})}=99.6 \%\) and \({F}_{\uparrow \uparrow }^{({\rm{p}})}=99.2 \%\) respectively. Thus, we find an average fidelity for the projectionSWAP of \({\bar{F}}_{{\rm{SWAP}}}^{({\rm{p}})}=98 \%\). The spinpolarized input states are insensitive to errors due to noise in the drive field and pulse miscalibrations, but should be sensitive to spin relaxation. The 96.5% fidelity for antiparallel spin input states is, therefore, not likely limited by relaxation. This is expected, since \({T}_{1}\) is 134 ms (52 ms) for \({Q}_{3}\) (\({Q}_{4}\)), which gives an upper bound of 99.97% fidelity for implementing 60 SWAPs, each padded with a 100 ns idle. The likely source of errors for antiparallel spin states arises from timedependent fluctuations in the magnetic field gradients or exchange interaction, which lead to miscalibrations in the projectionSWAP gate.
The highfidelity of our projectionSWAP gates implies that this SPAM technique can be useful for much larger arrays than the DQD configuration studied here. Our current fidelity suggests we could shuttle a spin projection across a ninedot array^{26} with a fidelity of 85%. It is notable that while the SWAP gate leads to spin transport between adjacent dots, the electron wavefunctions remain localized on the dots and there is no charge transport. This is in contrast to typical spinshuttle experiments^{27} that physically move electrons, a process which can be complicated by lowlying valley states,^{28} spin–orbit coupling,^{29} or spinrelaxation hotspots.^{2} Because spin transport can be controlled using only barrier gates, no fast plunger gate control is necessary, which should enable the operation of devices having fixed charge configurations. This could be of particular interest in twodimensional arrays where for even small numbers of qubits, charge state control and readout of interior sites becomes unmanageable. Finally, the projectionSWAP is compatible with singlettriplet readout^{8,21,30} and cavitybased dispersive readout.^{31,32}
CoherentSWAP
With the projectionSWAP, we have shown that it is possible to transfer spin eigenstates oriented along the magnetic field axis. More generally, the ability to transfer arbitrary quantum information is crucial to operating multiqubit devices with limited connectivity. Therefore, having achieved a highfidelity projectionSWAP, we turn our attention to transferring product states oriented along arbitrary directions with the coherentSWAP, e.g.,\(({\alpha }_{1}\left\uparrow \right\rangle +{\beta }_{1}\left\downarrow \right\rangle )\otimes ({\alpha }_{2}\left\uparrow \right\rangle +{\beta }_{2}\left\downarrow \right\rangle )\to ({\alpha }_{2}\left\uparrow \right\rangle +{\beta }_{2}\left\downarrow \right\rangle )\otimes ({\alpha }_{1}\left\uparrow \right\rangle +{\beta }_{1}\left\downarrow \right\rangle )\). The coherentSWAP has additional calibration requirements outlined in the Methods section. These tuning requirements are also sufficient for performing SWAP gates on entangled states (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory). Here we realize a coherentSWAP in 302 ns, which can be made faster by superimposing a dc exchange pulse as shown in Supplementary Fig. 2 (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory).
To verify our calibration, we prepared \({Q}_{3}\) in a superposition state, and performed state tomography before and after applying a coherentSWAP. By measuring the \(x\), \(y\), and \(z\) spin projections, we are able to reconstruct the singlespin density matrices \({\rho }_{i}\) as plotted in Fig. 4a. The imaginary components of \({\rho }_{i}\) are shown in Supplementary Fig. 5 (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory). From these data, we can estimate the SWAP fidelity \(F(\rho )\) by comparing the output state to the targeted state \({\psi }_{{\rm{ideal}}}\) using \(F(\rho )=\left\langle {\psi }_{{\rm{ideal}}}\right\rho \left{\psi }_{{\rm{ideal}}}\right\rangle\) and \(\rho ={\rho }_{3}\otimes {\rho }_{4}\).^{33} When constructing the twoqubit density matrices, SPAM errors are subtracted out (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory).^{2,4} This analysis gives a state fidelity of \(F(\rho )=89 \%\). Because this technique only measures the fidelity of swapping one pair of input states, obtaining an average gate fidelity requires repeating the experiment for each possible input.
To measure the average SWAP fidelity, we turn to Clifford randomized benchmarking, which is insensitive to SPAM errors.^{5,6,12,34} In Clifford randomized benchmarking, quantum circuits consisting of \(N\) randomly chosen Clifford gates are applied to a qubit, and at the end of the sequence, the qubit is rotated into a known state. Any gate infidelity throughout the sequence leads to errors in the final state. The qubit is measured and the experiment is repeated varying \(N\). As \(N\) increases, integrated errors cause the qubit state to become mixed and the probability of measuring \({P}_{i,\uparrow }\) approaches 50%.
To avoid the extensive overhead associated with full twoqubit randomized benchmarking, which requires calibration of an entangling twoqubit gate in addition to the SWAP gate, we use a technique pioneered by Chen et al.^{35} to benchmark twoqubit gates using interleaved randomized benchmarking.^{6,12} We note that while there are no entangled states generated by our single qubit Clifford gates, entanglement generation exists as a decay channel. We first perform single qubit Clifford randomized benchmarking on \({Q}_{3}\) and \({Q}_{4}\) by measuring the spinup probability \({P}_{\left\uparrow \uparrow \right\rangle }\) as a function of sequence length \(N\). These data, shown in black in Fig. 4b, are acquired with \({Q}_{3}\) and \({Q}_{4}\) single qubit rotations implemented in parallel. We next repeat the randomized benchmarking experiment, this time interleaving \(coherent\)SWAP gates after each set of parallel single qubit Clifford gates on \({Q}_{3}\) and \({Q}_{4}\). These results are plotted in red in Fig. 4b. The decays are fit to \({P}_{\left\uparrow \uparrow \right\rangle }={A}_{0}{p}_{c}^{m}+C\) where \({A}_{0}\) is the measurement visibility, \(C\) is the dark count, and \({p}_{c}\) is a decay parameter.^{12} This fit yields a decay parameter \({p}_{c}\) = 0.843 for the reference curve and \({\bar{p}}_{c}\) = 0.665 for the interleaved curve. By comparing these decay parameters we can extract an average coherentSWAP fidelity of \({\bar{F}}_{{\rm{SWAP}}}^{({\rm{c}})}=13/4(1\bar{{p}_{c}}/{p}_{c})=84 \%\), which is in good agreement with our estimate from state tomography.^{12}
Discussion
In conclusion, we have demonstrated a resonant SWAP gate that can be used for coherent spin transport and highfidelity state preparation and readout in an array of quantum dot spin qubits. We measure an average projectionSWAP gate fidelity of \({\bar{F}}_{{\rm{SWAP}}}^{({\rm{p}})}=98 \%\) when transferring eigenstates with a 100 ns gate time. We further show that a coherentSWAP gate can be used to transfer arbitrary twoqubit states between spins with an average fidelity of \({\bar{F}}_{{\rm{SWAP}}}^{({\rm{c}})}=84 \%\) in ~300 ns as measured using interleaved randomized benchmarking. By implementing automatic calibration and feedback,^{5} we should be able to significantly improve this fidelity. SWAP gates are an important building block in any quantum processor with limited qubittoqubit connectivity and are necessary to unlock the full capabilities of the multiqubit devices currently being fabricated in Si/SiGe.^{13,26} This robust implementation of a resonant SWAP gate promises to enable beyond nearestneighbor operation in quantum dot arrays, which is necessary for quantum information processing with more than two qubits.
Methods
Beyond the calibration required for the projectionSWAP, there are three additional constraints that must be satisfied to achieve highfidelity coherentSWAP gates. First, the resonant SWAP pulse must remain phase coherent with the qubits in their doubly rotating reference frame between calibrations (i.e. for hours). Second, because of the constraint that the exchange interaction is always positive, the timeaveraged exchange pulse necessarily has some static component, which leads to evolution under an Ising interaction as discussed in Supplementary Discussions I–III (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory). These Ising phases must be calibrated out. Finally, voltage pulses on any gate generally displace both electrons by some small amount. This movement induces phase shifts in both qubits, since they are located in a large magnetic field gradient. These phase shifts must be compensated for.
To satisfy these additional tuning requirements, we first ensure that our RF exchange pulse remains phase coherent. Each qubit’s reference frame is defined by the microwave signal generator controlling it, so by mixing together the local oscillators of these signal generators, we obtain a beat frequency that is phase locked to the doubly rotating twoqubit reference frame. We then amplitude modulate this signal to generate our exchange pulses. A detailed schematic is shown in Supplementary Fig. 6 (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory).
To calibrate for the single and twoqubit Ising phases, we use state tomography on both qubits before and after applying a SWAP gate. In these measurements, we vary the input states to distinguish between errors caused by twoqubit Ising phases, and the single qubit phase shifts. We choose a SWAP time and amplitude such that the Ising phases cancel out, which for this particular configuration occurs for a 302 ns SWAP gate. The single qubit phase shifts were measured to be 180° for \({Q}_{3}\) and 140° for \({Q}_{4}\). By superimposing the SWAP pulse with a dc exchange pulse, one can compensate for the Ising phases at arbitrary SWAP lengths, leading to faster operation (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory).
Data availability
The data supporting the findings of this study are available within the paper and its Supplementary Information (see Supplemental Material at https://doi.org/10.1038/s4153401902250 for additional gate tuneup data and theory). The data are also available from the authors upon reasonable request.
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Acknowledgements
Funded by Army Research Office grant no. W911NF1510149, DARPA grant no. D18AC0025, and the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant GBMF4535. Devices were fabricated in the Princeton University Quantum Device Nanofabrication Laboratory. The authors acknowledge the use of the Princeton University Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation MRSEC program (DMR1420541).
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A.J.S., M.J.G., and J.R.P. designed and planned the experiments, A.J.S. fabricated the device and performed the measurements, M.J.G. provided theory support, L.F.E. and M.B. provided the isotopically enriched heterostructure. A.J.S., M.J.G., and J.R.P. wrote the manuscript with input from all the authors.
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Sigillito, A.J., Gullans, M.J., Edge, L.F. et al. Coherent transfer of quantum information in a silicon double quantum dot using resonant SWAP gates. npj Quantum Inf 5, 110 (2019). https://doi.org/10.1038/s4153401902250
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DOI: https://doi.org/10.1038/s4153401902250
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