Abstract
Scalable quantum processors require tunable twoqubit gates that are fast, coherent and longrange. The Heisenberg exchange interaction offers fast and coherent couplings for spin qubits, but is intrinsically shortranged. Here, we demonstrate that its range can be increased by employing a multielectron quantum dot as a mediator, while preserving speed and coherence of the resulting spinspin coupling. We do this by placing a large quantum dot with 50–100 electrons between a pair of twoelectron double quantum dots that can be operated and measured simultaneously. Twospin correlations identify coherent spinexchange processes across the multielectron quantum dot. We further show that different physical regimes of the mediated exchange interaction allow a reduced susceptibility to charge noise at sweet spots, as well as positive and negative coupling strengths up to several gigahertz. These properties make multielectron dots attractive as scalable, voltagecontrolled coherent coupling elements.
Introduction
The Heisenberg exchange interaction between neighboring quantum dots allows precise voltage control over spin dynamics, due to the ability to precisely control the overlap of orbital wavefunctions by gate electrodes. This allows the study of fundamental electronic phenomena^{1,2,3,4} and finds applications in quantum information processing^{5}. Although spinbased quantum circuits based on shortrange exchange interactions are possible^{6,7}, the development of scalable, longerrange coupling schemes constitutes a critical challenge within the spin–qubit community. Approaches based on capacitative coupling^{8,9} and cavitymediated interactions^{10,11,12} effectively couple spin qubits^{13,14} to the charge degree of freedom^{15,16}, making them susceptible to electricallyinduced decoherence. The alternative is to extend the range of the Heisenberg exchange interaction by means of a quantum mediator^{17,18,19,20}.
Here, we show that a multielectron quantum dot with 50–100 electrons serves as an excellent mediator, preserving speed and coherence of the resulting spin–spin coupling while providing several functionalities that are of practical importance. These include speed (mediated twoqubit rates up to several gigahertz), distance (of order of a micrometer), voltage control, possibility of sweet spot operation^{21,22} (reducing susceptibility to charge noise), and reversal of the interaction sign (useful for dynamical decoupling from noise)^{4,23,24}.
Results
Implementation of longrange coupling
We implement longrange exchange coupling mediated by a multielectron quantum dot in a linear array of five quantum dots, as shown in Fig. 1a. The quintuple dot is defined in a GaAs twodimensional electron gas by means of electrostatic gate electrodes deposited on top of the heterostructure (see Methods section). The middle dot is populated by a large even number of electrons, between 50 and 100 as estimated from the lithographic size of the device and the density of the twodimensional electron gas. Its ground state is chosen to be spinless as described in ref. ^{4}. Two twoelectron double dots are tunnelcoupled^{4} on opposing sides of the large middle dot and are each initialized and read out using standard techniques for singlet–triplet qubits^{5,25}.
The exchange interaction is induced by a sequence of submicrosecond voltage pulses applied to the bluecolored gates in Fig. 1a, realizing the following steps (cf. Fig. 1b and further details about the pulse sequence in Supplementary Note 1). First, the outer dots are each populated by a pair of electrons. This initializes each double dot in the spin singlet state, \(S^{{\mathrm{L/R}}}\rangle = ( \uparrow \downarrow \rangle   \downarrow \uparrow \rangle )/\sqrt 2\), where arrows indicate the spin state of the two electrons and the superscript L/R indicates the left and right double dot. Then, the electron pairs are rapidly separated within each double dot. Since electron wave functions effectively stop overlapping, this pulse turns off the exchange interaction within each double dot, allowing the outer dots to each store one reference spin. In the third step, V_{M} is temporarily increased by ε_{M}, while negative (compensation) pulses are applied to all other gates (see Supplementary Note 1). This induces an exchange interaction between the inner oneelectron dots mediated by the large dot. After the interaction time τ the exchangeinducing pulse is switched off. Subsequently, spintocharge conversion is used to read out the relative spin alignment within each double dot^{5}, independently and with singleshot fidelity (see Methods section).
The result of such a spinexchange pulse sequence is shown in Fig. 2a. In the two panels, we plot the fraction of singlet outcomes (P_{S}) for each double dot as a function of τ and ε_{M}. Oscillations in P_{S} witness exchangedriven flipflop processes between the two spins located on the inner quantum dots. The oscillation frequency increases for larger values of ε_{M}. Consistent with complementary spinleakage spectroscopy (see Supplementary Note 2), this indicates that positive pulses on gate V_{M} lower the multielectron dot levels towards resonance with the inner dots, thereby increasing the rate of spinexchange processes mediated by the multielectron dot.
Correlated measurements
Next, we demonstrate correlations between measurement outcomes for the left and right double dot. For fixed interaction time τ = 2 ns, the demodulated voltage signals for the left and right sensor (V_{rf,L} and V_{rf,R}) exhibit correlations that oscillate with the amplitude of the applied pulse (Fig. 2b), confirming the nonlocal mechanism underlying panel 2a (in Fig. 2b, the exchangeinducing pulses parametrized by \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\) are defined similarly to ε_{M}pulses in Fig. 2a, but employ more sophisticated crosscompensation pulses, as described in Supplementary Note 1). From these histograms, we extract the joint probabilities of detecting a singlet (S) or triplet (T) for the two double dots as a function of \(\varepsilon _{\mathrm{M}}^{{\mathrm{cc}}}\) (see Supplementary Movie 1 for animated histograms). Figure 2c clearly shows anticorrelated probabilities for detecting SS and TT, whereas the probabilities of ST and TS are small and nearly constant. The joint probabilities were extracted by fitting the histograms with four Gaussians and correcting for doubledot relaxation during the measurement pulse (see Supplementary Note 3).
The oscillatory behavior of joint probabilities results from the precession between the initialized state S^{L}〉S^{R}〉 and the maximally entangled state \(\frac{1}{2}(S^{\mathrm{L}}\rangle S^{\mathrm{R}}\rangle  T_0^{\mathrm{L}}\rangle T_0^{\mathrm{R}}\rangle + T_ + ^{\mathrm{L}}\rangle T_  ^{\mathrm{R}}\rangle + T_  ^{\mathrm{L}}\rangle T_ + ^{\mathrm{R}}\rangle )\). Here, the two kets denote the state of the left and right double dot, respectively, and the spin triplet states are labeled according to the standard convention, \((T_0\rangle = ( \uparrow \downarrow \rangle +  \downarrow \uparrow \rangle )/\sqrt 2 ,T_ + \rangle =  \uparrow \uparrow \rangle ,T_  \rangle =  \downarrow \downarrow \rangle )\). The coefficients associated with this entangled state explain the visibility in our measurement basis. For example, the maximum expected probability for TT is 75% in the case of perfect readout, nearly matched by the observed maxima in Fig. 2c. The observed visibility for SS and TT is further reduced by residual counts of ST and TS. We attribute this background to unintentional dynamics of the reference spins in the outer dots, arising from decoherence due to their coupling to the nuclear spin bath associated with GaAs^{26}, and from the finite rise time of the voltage pulses.
In this experiment, we can exclude capacitive coupling as a mechanism leading to the observed oscillations, as it is small and manifests itself as a controlled phase gate between singlet–triplet qubits^{8,9}, not as a SWAP gate between single spins.
Physical regimes of exchange interaction
In Fig. 3, we identify different mechanisms of the exchange interaction mediated by the multielectron quantum dot. For that purpose, we define a new gate voltage parameter, \(\varepsilon = (V_{{\mathrm{L2}}}  V_{{\mathrm{R1}}})/\sqrt 2 + C\) (where C is a constant; see Supplementary Note 1), which controls the relative detuning between the two inner dots, and plot the readout probabilities P_{S}(ε, ε_{M}). Since τ = 6 ns is fixed for these measurements, each fringe in Fig. 3b corresponds to points of equal exchange energy J, while the density of fringes represents the gradient of J (see Supplementary Note 8 for a discussion of finiterisetime effects). The observed exchange strength rapidly increases when ε_{M} > 20 mV, especially for ε ≈ 0, resulting in a high density of fringes that is blurred by a combination of aliasing and decoherence. For finite ε, the exchange increases more slowly, resulting in a pattern that is approximately symmetric with respect to ε. While Fig. 3 presents measurements obtained from the right double dot, identical patterns are observed from the left double dot (see Supplementary Note 5). This is not the case when the occupancy of the multielectron dot is increased by one electron, as discussed in Supplementary Note 6.
The observed pattern can be understood by monitoring the charge distribution during the interaction step (Supplementary Note 4), and overlaying the fringes in Fig. 3b with the observed charge transitions (dashed lines). Consistent with simulations from the Hubbard model discussed below, shown in Fig. 3e, f, we identify each region with a different electron configuration, as illustrated by dots in Fig. 3a. In region I, the inner dots remain singly occupied, and the multielectron dot keeps its initial charge state. This corresponds to an indirect exchange interaction, where virtual tunneling through the multielectron quantum dot mediates superexchange^{19}. In regions IIa and IIb, one of the electrons has relocated onto the multielectron dot, forming an effective spin1/2 manybody state which directly exchangecouples to the other electron spin. The mirror symmetry of IIa and IIb with respect to ε^{*} = 0 reflects the left–right symmetry of the device, with minor deviations in the experimental data arising from a slight inequality in the tunneling barriers between the multielectron dot and the inner dots. In region III, the chemical potential of the multielectron dot is sufficiently low such that both electrons relocate onto the multielectron dot. Depending on their relative spin alignment, singletlike or tripletlike, both electrons occupy either the lowest orbital, or the lowest and second lowest orbital, respectively. The energy difference between these spin configurations sets the coupling strength of this (rapid) onsite exchange interaction. It is related to two mesoscopic parameters, namely the singleparticle level spacing of the two orbitals, and the spin correlation energy^{4,27,28}.
The different mechanisms of the exchange interaction can also be distinguished by reducing the tunnel couplings between the multielectron and the inner dots (Fig. 3c, d). We observe P_{S} ≈ 1 throughout region I, indicating that the reduced coupling effectively turns off the indirect exchange regime. Region I now appears separated by a sharp boundary, i.e., charge transition, from the direct exchange regions IIa,b. Similarly, in Fig. 3c, a sharp boundary between clear and blurred oscillation fringes is observed, indicating the transition between direct and onsite exchange regimes. Upon further reduction of tunnel coupling (Fig. 3d), an arc pattern is observed in region III. This behavior results from a sweet spot and a sign reversal of the exchange interaction^{4,23}, as argued below.
Model and sweetspot behavior
To verify all four regimes, we evaluate a Hubbard model of the two inner quantum dots coupled to the multielectron quantum dot (see level structure in Fig. 3a and Supplementary Note 7). Using realistic parameters, this model qualitatively reproduces data in Fig. 3b, including the fringe pattern and the charge stability diagram (Fig. 3e, f).
Quantitative insight into the fast dynamics of onsite exchange can be obtained by reducing τ to 2 ns. This circumvents blurring and aliasing effects, revealing a characteristic arc pattern at the transition between direct and onsite exchange regimes (Fig. 4a). This pattern (similar to the one observed in Fig. 3d) is in fact a fingerprint of the exchange profile J(ε, ε_{M})^{22}: Retaking any pixel, say along a cut at ε = −8 mV (blue triangle in Fig. 3b), as a function of τ results in oscillations with frequency f = J/h^{21}. Extracting f for the cut shown in Fig. 4b reveals a nonmonotonic behavior of the exchange coupling with respect to ε_{M} (Fig. 4d). The presence of a maximum in frequency followed by a zero crossing is similar to exchange profiles studied in refs. ^{4,23}, and arises if direct exchange (which depends on orbitalspecific tunnel matrix elements) competes with onsite exchange (which depends on electron correlation effects and, for relatively small orbital level spacing, can be negative). Accordingly, to qualitatively reproduce the chevron pattern of Fig. 4b, we must include two unoccupied orbitals of the multielectron quantum dot^{4,24}, as well as a finite rise time of the applied voltage pulses (see Methods and Supplementary Note 7). The results of such a simulation are shown in Fig. 4d and are in good agreement with experimental data.
Furthermore, the observed visibility of oscillations in Fig. 4b depends on ε_{M}, which we associate with an enhancement of fidelity in two operating regimes. First, for large values of ε_{M}, the onsite exchange energy is set by the (mesoscopic) level spacing of the dot, which to lowest order is insensitive to pulse amplitudes. This regime is akin to the noiseinsensitive regimes noted in refs. ^{14,29} and exploited by the threeelectron doubledot hybrid qubit^{30,31}. Second, highfidelity oscillations appear along the curved chevron pattern, suggesting that the local extremum in the exchange strength provides insensitivity^{21,22} to fluctuations in ε_{M}. For this tuning of the device, the oscillation frequency in both noiseinsensitive regimes exceeds 5 GHz, making it challenging to perform small angle rotations using conventional pulse generators. However, by decreasing the tunnel couplings between the multielectron dot and the inner dots, the operating speed at the sweet spot can be reduced as desired (down to 1 GHz as demonstrated in ref. ^{23}).
Discussion
The significant coupling strength suggests realistic application in quantum information processing where fast spin exchange is needed between nonnearest neighbors. An alternative approach for coupling distant spin qubits (based on superconducting cavities and spin–photon coupling) is expected to be orders of magnitude slower, although no photonmediated spin–spin coupling has been demonstrated to date (for comparison, strong spin–photon exchange itself has been observed, and is rather slow^{10,11}).
Phenomenologically, the most striking difference between indirect exchange in our device and similar manifestations in a doublyoccupied triple dot^{17} is its occurrence in a charge configuration that is left–right symmetric (see Supplementary Note 9). During this indirect exchange, two singletcorrelated electron pairs are coherently linked over a linear array of five dots, making this the largest coherent quantum dot array to date. Microscopically, we speculate that the underlying mechanism can be viewed as a smallsystem manifestation of the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, which has no such counterpart in the direct and onsite regimes^{19}.
An interesting next step building upon this demonstration is to employ a multielectron quantum dot of larger dimensions, with multiple singleelectron quantum dots around its perimeter. This will allow coherent coupling of arbitrary pairs of electrons, and may lead to a programmable hardware architecture in which qubit–qubit connectivities can be reconfigured in situ to best serve the specific computational tasks. Increasing the coupler size has additional advantages, such as reducing the onsite exchange energy which would enable performing highfidelity, smallangle rotations. Another direction is the implementation of this coupling scheme in silicon nanostructures, mitigating decoherence effects arising from the nuclear spin bath. Our demonstration of coherently swapping spin pairs across the multielectron quantum dot suggests that shuttling of individual electrons^{32} through the multielectron quantum dot should also be feasible. Combinations of these achievements will open many paths for scaling quantumdotbased qubit circuits.
Methods
Sample preparation
The array of quantum dots is defined in a highmobility (230 m^{2} V^{−1} s^{−1} at 0.3 Kelvin) twodimensional electron gas (density 2.5 × 10^{15} m^{−2}) located 57 nm below the surface of a GaAs/AlGaAs heterostructure, by means of electrostatic gate electrodes deposited on top of the heterostructure^{4}. A 10nmthick layer of HfO_{2} is deposited on top of the heterostructure, prior to patterning the gold electrodes by electronbeam and liftoff lithography. A top view of the gate electrodes is shown in Fig. 1a. A detailed characterization of this device chip can be found in ref. ^{4}.
Readout
Within each double dot, spintocharge conversion is used to read out the relative spin alignment within each double dot^{5}. This is done at the end of each waveform cycle (cf. Supplementary Figure 1 in Supplementary Note 1). Specifically, a frequencymultiplexed measurement pulse reflected off two proximal radiofrequency quantumdotbased charge sensors^{25} allows us to distinguish between singlet and triplet states of each double dot, independently and with singleshot fidelity. To reduce errors arising from slow drifts in the demodulated sensor voltages (V_{rf,L}, V_{rf,R}), within each waveform cycle we acquire sensor voltages just after initialization of the singletsinglet state (\(V_{{\mathrm{rf,L}}}^i\) and \(V_{{\mathrm{rf,R}}}^i\)) and after the actual interaction step (\(V_{{\mathrm{rf,L}}}^f\) and \(V_{{\mathrm{rf,R}}}^f\)). These two measurements correspond to the “Reference measurement” and “measurement” segment indicated in Supplementary Figure 1. By defining the measurement outcomes relative to the reference outcomes, the adverse effects of slowly drifting sensor signals are effectively removed: \(V_{{\mathrm{rf,L}}} \equiv V_{{\mathrm{rf,L}}}^f  V_{{\mathrm{rf,L}}}^i\) and \(V_{{\mathrm{rf,R}}} \equiv V_{{\mathrm{rf,R}}}^f  V_{{\mathrm{rf,R}}}^i\).
Implementation of exchange pulses with subnanosecond resolution
To achieve subnanosecond resolution of the exchange pulse, we interfered two nominally canceling signals generated by two arbitrary waveform generator channels, and applied the combined signal to the multielectrondot plunger gate V_{M}. Specifically, we set the two channels to output a square waveform of identical duration and amplitude, but with opposite polarity, and combine them using an inverted power splitter. The pulse period is set to the repetition time of the intended pulse sequence, the rising edge of the pulse is set to the beginning of the intended exchange pulse, while the falling edge happens at the beginning of the doubledot initialization step. By finely adjusting the channel skew of the arbitrary waveform generator, positive or negative V_{M} pulses can be generated with subnanosecond control. While this method allows to overcome the limitations of the waveform generator’s temporal resolution of 1.2 GS s^{−1} (Tektronix AWG 5014C), the effective voltage pulse reaching the gate electrodes is still constrained by the 0.8 ns pulse rise time in our dilution refrigerator, resulting in distortion effects in Figs. 2–4 (cf. Supplementary Note 8).
Data availability
The datasets generated and analyzed during the current study are available from the corresponding author (F.K.) upon reasonable request.
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Acknowledgements
This work was supported by the Army Research Office, the Innovation Fund Denmark, the Villum Foundation, the Danish National Research Foundation, and the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE170100009. Work at Purdue was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DESC0006671. Additional support from Nokia Bell Labs for the GaAs MBE effort is also gratefully acknowledged.
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Contributions
S.F., G.C.G. and M.J.M. developed the GaAs heterostructures. P.D.N. fabricated the devices. F.K.M. and F.M. carried out the measurements with input from C.M.M. and F.K. Data analysis was done by F.K.M. and F.M., and theoretical models and simulations were developed by T.B.S., S.D.B., and A.C.D. All authors contributed to interpreting the data. The manuscript was written by F.K.M., F.M. and F.K. with suggestions from all other authors.
Corresponding author
Correspondence to Ferdinand Kuemmeth.
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Malinowski, F.K., Martins, F., Smith, T.B. et al. Fast spin exchange across a multielectron mediator. Nat Commun 10, 1196 (2019). https://doi.org/10.1038/s4146701909194x
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