Abstract
Quantum coherence of superposed states, especially of entangled states, is indispensable for many quantum technologies. However, it is vulnerable to environmental noises, posing a fundamental challenge in solidstate systems including spin qubits. Here we show a scheme of entanglement engineering where pure dephasing assists the generation of quantum entanglement at distant sites in a chain of electron spins confined in semiconductor quantum dots. One party of an entangled spin pair, prepared at a single site, is transferred to the next site and then adiabatically swapped with a third spin using a transition across a multilevel avoided crossing. This process is accelerated by the noiseinduced dephasing through a variant of the quantum Zeno effect, without sacrificing the coherence of the entangled state. Our finding brings insight into the spin dynamics in open quantum systems coupled to noisy environments, opening an avenue to quantum state manipulation utilizing decoherence effects.
Introduction
One important goal of quantum technologies is to utilize entangled states in isolated, welldefined systems in a controllable manner. Decoherence, which scrambles the correlation between entangled parties through the interaction with the environment, is a major enemy of those quantum technologies, including quantum computation. In semiconductor quantum dot (QD) devices, potential building blocks of spinbased quantum computers, a great deal of effort have been made to mitigate the decoherence by engineering^{1,2}, controlling^{3}, and measuring^{4,5} the environmental noise sources.
The decoherence effect is particularly significant in entangling gate operations for spin qubits because they are implemented by electrically tuning exchange coupling, which makes these qubits sensitive to charge noise^{6}. To prepare an entangled state between nonnearestneighbor spins, for instance, one could initialize a spinsinglet state in a double quantum dot and repeat the SWAP operations^{7} between one of the two spins and the nextnearestneighbor spins. For fast and highfidelity control, the interdot exchange coupling J has to be sufficiently larger than the Zeeman energy gradient ΔB, while a large ΔB is also favorable for addressable singlespin control^{8,9,10}. Repeating the SWAP operations with large J is, however, practically difficult because it demands precise electrical control of subnanosecond pulse trains and the coherence time is decreased by the enhanced sensitivity to charge noise. In the opposite limit of \(J \ll \Delta B\), the qubits are insensitive to charge noise and therefore an entangled state, once created, evolves coherently until it is dephased by inhomogeneous broadening of the Zeeman energy. Despite difficulties in engineering entanglement manipulation, quantum entanglement is believed to play essential roles even in systems coupled to noisy environments, including certain biological organisms^{11}. Indeed, it was shown that decoherence can be used as a resource to stabilize entanglement in an artificial system^{12}.
Here we consider shuttling an entangled state to distant sites through an artificial spin chain (Fig. 1a). By using a simple, linear ramp of the detuning energy in an array of quantum dots with a magnetic field gradient (Fig. 1b), a typical setup for addressable singlespin control, the system undergoes adiabatic transitions across multilevel energy crossings as shown in Fig. 1c. At each energy crossing, one party of an entangled spin pair is swapped with the next spin. Repeating a similar process one after another yields entanglement between distant sites without direct interaction. We find that the efficiency of the adiabatic swap is significantly enhanced by pure dephasing—decoherence with no energy dissipation, suppressing nonadiabatic transitions across those energy crossings. Counterintuitively, the coherence of the entangled state is preserved during this process. This scheme is generally applicable to an array of more than three quantum dots. We demonstrate this scheme in a gatedefined GaAs/AlGaAs triple quantum dot (TQD) with thoroughly characterized energy levels. Formation of an entangled state between sites a and b is demonstrated by observing the coherent evolution of a superposed state (singlettriplet oscillation), \(\left \psi \right\rangle = \alpha \left { \downarrow _a \uparrow _b} \right\rangle + e^{  i\Delta B_{ab}t/\hbar }\beta \left { \uparrow _a \downarrow _b} \right\rangle\), where the two entangled sites a and b can be identified by the oscillation frequency determined by the Zeeman energy difference, \(\Delta B_{ab}/h\). Numerical simulations show that the adiabatic spin swap is assisted by strong dephasing noise, which can be interpreted as a manifestation of the quantum Zeno effect.
Results
Generation and detection of correlated spin states
The charge configuration of the TQD is controlled by the gate biases applied to plunger gates and probed by a nearby rfQD charge sensor^{13}. With the number of electrons in each QD, N_{ i }, set to (N_{1}N_{2}N_{3}) = (111), voltage pulses are superposed to the gate biases to control the energy offset between charge configurations (102) and (201) along the line shown in Fig. 2c. This allows us to control the detuning energy ε between the left and right QDs while leaving the energy of the (111) configuration unchanged. A top cobalt micromagnet induces an inhomogeneous Zeeman field^{8,9,10} along the externally applied inplane magnetic field B_{ext}. By design, the transverse field component induced by the micromagnet is much smaller. The Zeeman energy difference ΔB_{ ij } between QD_{ i } and QD_{ j } (i,j = 1, 2, 3) splits the degenerate threespin states with the same z component of total spin to discrete levels E_{ k } (labeled from lower to higher energies at ε = 0 as shown in Fig. 2d). When the exchange energy J_{ ij } is negligible (\( {\Delta B_{ij}/J_{ij}}  \gg 1\)), the threespin eigenstates are described by \(\left {\sigma _1\sigma _2\sigma _3} \right\rangle = \left {\sigma _1} \right\rangle \otimes \left {\sigma _2} \right\rangle \otimes \left {\sigma _3} \right\rangle\) (\(\sigma _i = \uparrow , \downarrow\)) rather than the quadruplet and doublets^{14,15,16}, allowing us to access individual spin states^{17}.
An entangled spin pair is locally prepared by initializing the system in (102), where two electrons in QD3 occupy the singlet ground state, S_{3}. The electron in QD1 is left uninitialized and its spin state is thermally populated with the Boltzmann distribution. Thus the system is initially in a mixed state described by the density matrix, \(\rho _0 = r\left { \uparrow _1{\mathrm{S}}_3} \right\rangle \left\langle { \uparrow _1{\mathrm{S}}_3} \right + \left( {1  r} \right)\left { \downarrow _1{\mathrm{S}}_3} \right\rangle \left\langle { \downarrow _1{\mathrm{S}}_3} \right\), with r ≈ 0.7 resulting from the electron temperature, T_{e} = 240 mK. The spin pair is then split to QD2 and QD3 by rapidly displacing ε to zero in (111) (Methods section). When the displacement is slow enough compared to the interdot tunnel rate t_{R}, but rapid enough against ΔB_{23}, the spin state is unchanged and \(\left {{\mathrm{S}}_3} \right\rangle\) is loaded into a spin singlet between QD2 and QD3, \(\left {{\mathrm{S}}_{23}} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left { \uparrow _2 \downarrow _3} \right\rangle  \left { \downarrow _2 \uparrow _3} \right\rangle } \right)\) (as is the case for double QDs when QD1 is neglected). The loaded state starts to oscillate with time t_{evolve} between \(\left {{\mathrm{S}}_{23}} \right\rangle\) and \(\left {{\mathrm{T}}_{23}} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left { \uparrow _2 \downarrow _3} \right\rangle + \left { \downarrow _2 \uparrow _3} \right\rangle } \right)\) (spin triplet) as shown in Fig. 3b, reflecting the relative phase evolution between \(\left { \downarrow _2 \uparrow _3} \right\rangle\) and \(\left { \uparrow _2 \downarrow _3} \right\rangle\) due to the energy difference ΔB_{23}/h = 45 MHz. The singlet fraction in the final state is read out by unloading S_{23} back to S_{3} in the reverse process. Hereafter, the probability of finding the singlet final state in (102) is denoted “singletreturn probability”, P_{S}.
Transfer of entangled states to a distant site
The entanglement created in QD2 and QD3 is then transferred to the distant spin pair in QD1 and QD3 by pulsing ε more negatively. A distinct feature of the coherent oscillation between S_{13} and T_{13} is visible in the region \(\varepsilon \ll \varepsilon _{\mathrm{L}}\), where the oscillation frequency is governed by the energy gap ΔB_{13} = ΔB_{12} + ΔB_{23} (red data points in Fig. 3b). This frequency is clearly distinguished from that of the S_{23}–T_{23} precession governed by ΔB_{23}. The generation of this distant entanglement could be understood by the adiabatic transition through a multilevel avoided crossing depicted in Fig. 2d. When ε is swept from ε = 0 to \(\varepsilon \ll \varepsilon _{\mathrm{L}}\), \(\left { \uparrow _1 \downarrow _2 \uparrow _3} \right\rangle\) is adiabatically loaded into \(\left { \downarrow _1 \uparrow _2 \uparrow _3} \right\rangle\), following the eigenenergy marked as E_{2}. This process adiabatically swaps the spins in QD1 and QD2, transferring \(\left { \uparrow _1{\mathrm{S}}_{23}} \right\rangle\) to \(\left { \uparrow _2} \right\rangle \otimes \left {{\mathrm{S}}_{13}} \right\rangle = \frac{1}{{\sqrt 2 }}\left( {\left { \uparrow _1 \uparrow _2 \downarrow _3} \right\rangle  \left { \downarrow _1 \uparrow _2 \uparrow _3} \right\rangle } \right)\). On the other hand, the fraction of \(\sigma _1 = \downarrow\) in ρ_{0} is transferred to \(\left { \downarrow _1{\mathrm{S}}_{23}} \right\rangle\) at ε = 0 and then to \(\frac{1}{{\sqrt 2 }}\left( {\left {{\mathrm{S}}_1 \downarrow _3} \right\rangle  \left { \downarrow _1 \downarrow _2 \uparrow _3} \right\rangle } \right)\) at ε < ε_{L} [as \(\left { \downarrow _1 \uparrow _2 \downarrow _3} \right\rangle\) is unloaded to \(\left {{\mathrm{S}}_1 \downarrow _3} \right\rangle\), see the upper panel of Fig. 2d]. Since this state is a superposition of different charge sectors in (111) and (201), it decoheres immediately^{6} and makes no contribution to the observed oscillations. However, the quantitative analysis presented below reveals that this naive interpretation is insufficient; as shown by the gray line in Fig. 3b obtained from a numerical simulation, only the S_{23}–T_{23} precession with frequency ΔB_{23}/h would be observed if strong dephasing noise were absent. This is because the coherent Landau–Zener transition nonadiabatically brings \(\left { \uparrow _1 \downarrow _2 \uparrow _3} \right\rangle\) to the identical spin state in the energy branch marked as E_{8}, as the relevant energy splitting, \(\left( {2t_{\mathrm{L}}/h} \right)^2 \approx 0.4\,{\mathrm{GHz}}^2\), is much smaller than the Landau–Zener velocity determined by the pulsing of ε, \(v_{{\mathrm{LZ}}}/h \approx 60\,{\mathrm{GHz}}^2\).
Threespin state spectroscopy
To simulate the coherent dynamics of the system, we extract the system energy diagram shown in Fig. 2d by measuring^{2,18} coherent oscillations of P_{S} at various values of ε. Figure 4a shows that the oscillation frequency changes with ε, representing the energy gaps between two pairs of superposed spin states (\(\left {E_3  E_2} \right\) and \(\left {E_5  E_4} \right\) marked by green arrows in Fig. 2d). The energy spectrum is mapped out by the 1D Fourier transform of this data as shown in Fig. 4b. A singlespectral peak at ɛ ≳ 0 represents that the energy gaps are almost unchanged around ε ≈ 0, being solely given by ΔB_{23}/h = 45 MHz since J_{12} and J_{23} are quenched, and increase equally with J_{23} for more positive ε. On the other hand, the spectral lines split at ε < 0 as the energy gap increases (decreases) for \(\sigma _1 = \uparrow\) (\(\downarrow\)) due to J_{12}. Fitting the two spectral lines with the model calculations (red solid and blue dashed lines in Fig. 4b) allows us to extract the threespin energy diagram without arbitrary parameters.
Although the observed oscillations are assigned to the coherent evolution between the energy levels discussed above, finite state leakage to other levels also happens. The relevant dynamics has been studied in double quantum dots^{19,20}. First, the singlet S_{3} prepared in QD3 may fail to tunnel to QD2 during the detuning sweep and the system stays in the (102) charge configuration (E_{10} or E_{11} energy branches). This state does not contribute to the coherent oscillations because the coherence with the states in the (111) charge sector is lost in a much shorter timescale. It contributes, however, to the increase of the mean value of P_{S}. Second, S_{3} may be loaded to a spinpolarized triplet state \({\mathrm{T}}_{ + 23} = \left { \uparrow _2 \uparrow _3} \right\rangle\) when the detuning crosses the energy resonance point. The position of the S_{3}–T_{+23} resonance is identified by measuring the S_{3}–T_{+23} mixing caused by the transverse magnetic field gradient. We find that this mixing rate is well below 10 MHz, as expected for the small transverse component (<10 mT) and the large external magnetic field (B_{ext} = 0.7 T). The state leakage to T_{+23} is therefore negligible when the detuning is swept past the resonance with the ramp time of t_{s} = 1.6 ns used here (Methods section). For this reason, the transverse field component is neglected in the simulation.
Analysis of the dephasing effect
The coupling of the spin system to pure dephasing noise in the environment results in fluctuations of the eigenenergies E_{ k }. Although the magnetic fluctuation of ΔB_{ ij } due to randomly oriented nuclear spins is significant in the GaAs material, its influence is pronounced at a rather longtimescale, leading to an ensemble phase averaging effect^{5,21}. Meanwhile, E_{ k } is also susceptible to the exchange noise in J_{ ij }, which is in turn susceptible to the charge noise in (ε–ε_{L,R}) and t_{L,R}. This effect is especially significant when the derivatives of E_{ k } with those parameters are large^{6} (red shaded region in Fig. 2d). Assuming that the noise sources are uncorrelated with each other, the fluctuation of E_{ k } can be represented by those of ε_{L,R} and t_{L,R} around ε ≈ ε_{L,R} without loss of generality. In the Markov approximation, this effect is described by the Lindblad operators
where \(\left k \right\rangle\) is the energy eigenstate of E_{ k }. The operators in this form lead to pure dephasing in a superposed state of \(\left k \right\rangle\) and \(\left l \right\rangle\) at the rate \(\Gamma _{kl} = \gamma _{\varepsilon _{{\mathrm{L,R}}}}\left( {\frac{{\partial E_{kl}}}{{\partial \varepsilon _{{\mathrm{L,R}}}}}} \right)^2 + \gamma _{t_{{\mathrm{L,R}}}}\left( {\frac{{\partial E_{kl}}}{{\partial t_{{\mathrm{L,R}}}}}} \right)^2\), with E_{ kl } = E_{ k } – E_{ l }. Based on these dephasing terms and the extracted energy diagram, we numerically simulate the coherent evolution of the system (Methods section). Figure 4c shows that the coherent oscillation of the distant entanglement, visible in Fig. 3b and in the region of ε < ε_{L} in Fig. 4a, is entirely reproduced only when the dephasing rates estimated from the experiment are taken into account. We notice, however, that the adiabatic transition from \(\left { \uparrow _1 \downarrow _2 \uparrow _3} \right\rangle\) (ε = 0) to \(\left { \downarrow _1 \uparrow _2 \uparrow _3} \right\rangle\) (\(\varepsilon < \varepsilon _{\mathrm{L}}\)) is not perfect in the simulation with our choice of the dephasing rates. This leads to a residual oscillatory component between \(\left { \uparrow _1 \downarrow _2 \uparrow _3} \right\rangle\) and \(\left { \uparrow _1 \uparrow _2 \downarrow _3} \right\rangle\) with frequency ΔB_{23}/h superposed onto the red line in Fig. 3b, which is not discernible in the experimental data. This discrepancy may be resolved by taking into account more accurate dephasing rates and noise spectrum. We thus demonstrate that the pure dephasing noise plays an essential role in performing the adiabatic spin swap in QD1 and QD2 required for generating distant entanglement.
Before discussing the interpretation of the result, we check the validity of our noise model by a charge dephasing measurement^{6} (Methods section). Figure 5a shows a typical damping curve of the exchange oscillation near ε = ε_{R}, being well characterized by an exponential decay rather than a Gaussian. This is consistent with the quasiwhite spectrum and the Markovian nature of the noise^{22}, which is here attributed to the Johnson–Nyquist noise on gate bias voltages (Methods section). The decay time extracted from the envelope is plotted versus ε in Fig. 5b. It is seen that the decay rate increases as ε approaches ε_{L,R} where the exchange noise dominates the magnetic fluctuation. The dephasing rate in this regime fits very well with Γ_{ kl } derived from Eq. 1, giving the values of \(\gamma _{\varepsilon _{{\mathrm{L,R}}}}\) and \(\gamma _{t_{{\mathrm{L,R}}}}\). This result clearly suggests that the dephasing rate overwhelms the relevant energy splittings, \(\Gamma _{kl} \gg t_{{\mathrm{L,R}}}/h\), such that the dephasing effect plays an essential role in the transition process^{23}.
Discussion
The effect of the dephasing noise can be interpreted as the manifestation of the quantum Zeno effect, also known as the ‘watchdog’ effect, which forces a system to be in an eigenstate by frequent projective measurements^{24}. More generally, the environmentinduced dephasing can be viewed as a continuous monitoring of the system by the environment^{23} as recently demonstrated experimentally^{25}. This effect tends to force Landau–Zener transitions^{26} to be adiabatic even when the Landau–Zener velocity is in the nonadiabatic regime. In this work, the transition between QD1 and QD2 would be nonadiabatic [\(\left( {2t_{\mathrm{L}}} \right)^2 \ll hv_{{\mathrm{LZ}}}\)] in the absence of the dephasing noise, so that spin entanglement fails to propagate to QD1 as shown in the right panel in Fig. 4c. However, the strong dephasing noise [\(\Gamma _{12},\Gamma _{28} \gg t_{\mathrm{L}}/h\)] keeps this process adiabatic, facilitating the adiabatic spin swap between QD1 and QD2. This effect is significantly enhanced around ε ≈ ε_{L}, where superposed states in E_{1} and E_{2} or in E_{2} and E_{8} are sensitive to the charge noise due to the large derivatives \(\left {\partial E_{kl}/\partial \varepsilon _{{\mathrm{L,R}}}} \right\) and \(\left {\partial E_{kl}/\partial t_{{\mathrm{L,R}}}} \right\). The resulting dephasing with rates Γ_{12} and Γ_{28} keeps the eigenstate of E_{2} staying in the instantaneous energy eigenstate throughout the passage across ε = ε_{L}. It is noteworthy that the coherent evolution of the generated entangled state is observed even in the presence of the strong dephasing noise utilized to generate it. This is because the entangled state is in the decoherencefree subspace^{27,28,29} spanned by E_{3} and E_{2}. This subspace is effectively decoupled from the identical noise sources thanks to much smaller \(\left {\partial E_{32}/\partial \varepsilon _{{\mathrm{L,R}}}} \right\) and \(\left {\partial E_{32}/\partial t_{{\mathrm{L,R}}}} \right\). We have thus demonstrated that, if properly used, the dephasing effect can enhance the manifestation of purely quantum mechanical properties in a noisy environment.
Finally, we comment on the applicability of the present technique to a larger array of quantum dot spins. The generation of a distantly entangled state by a singleadiabatic detuning pulse (Fig. 1) is technically much easier to implement than repeating the conventional SWAP operations by switching the exchange coupling nonadiabatically for calibrated durations. On the other hand, the detuning pulse must be sufficiently slow to realize the adiabatic limit, leading to slower control in a larger array. This requirement is partly alleviated by the use of dephasing noise as shown in the present work. However, the fidelity of the process is limited by the lack of controllability of the dephasing noise. We expect that control of dephasing noise by, e.g., external noise generators should allow faster and more reliable entanglement shuttling.
In conclusion, the noiseassisted transfer of spin correlations demonstrated in our experiment may open the possibility for arbitrary relocation of quantum entanglement and using its quantum nature in larger scale devices. The underlying physics may also be relevant to the possible manifestation of quantum entanglement in biological systems, such as photosynthetic lightharvesting complexes and avian chemical compasses^{11}. Our results suggest that the QD devices can be used as a powerful platform for quantum simulation of the open quantum systems coupled to noisy environments.
Methods
Measurement
The pulse sequences used in the experiment, illustrated in the Supplementary Fig. 1, were generated by a Tektronix AWG70002 arbitrary waveform generator operated at 1 GSa per s. The ouput waveform was lowpass filtered by MiniCircuits SBLP300+ to adjust the rise time of the rapid adiabatic pulse used in the experiment, as well as to filter out highfrequency noise. We find that the step response of the filter is well approximated by using the error function as \(\psi \left( t \right) = \frac{1}{2}\left[ {1 + {\mathrm{erf}}\left( {\frac{{t  t_0}}{{t_{\mathrm{s}}}}} \right)} \right]\) with t_{s} = 1.6 ns. This effect is taken into account in the numerical calculations described below. The pulse signals are fed to the sample through broadband coaxial cables in the dilution refrigerator with their −3 dB point at around 10 GHz. The signals are attenuated by 9 dB in total through cryogenic attenuators installed at each cold plate.
At the beginning of each pulse cycle, the spins in QD2 and QD3 are initialized to the doubly occupied singlet state S_{3}. For the entanglement measurement in Figs. 3b and 4a, this is achieved at point M_{0}, where the relaxation of the triplet state with antiparallel spins, T_{23}, is fast because of the rapid state mixing with S_{23} by ΔB_{23} and efficient phonon emission^{30}. The state \(\left {\sigma _1{\mathrm{S}}_3} \right\rangle\) is then rapidly loaded^{31,32} into (111) by setting the detuning to the target value ε at point O within the rise time of t_{s}. In this rapid adiabatic passage process, \(\left {\sigma _1{\mathrm{S}}_3} \right\rangle\) remains in the singlet state \(\left {\sigma _1{\mathrm{S}}_{23}} \right\rangle\) as far as J_{12} is negligible compared to ΔB_{12}. After the state evolution for t_{evolve}, the detuning is pulsed back for readout and \(\left {\sigma _1{\mathrm{S}}_{23}} \right\rangle\) returns to \(\left {\sigma _1{\mathrm{S}}_3} \right\rangle\) while all the other triplet components remain spinblocked. The readout is performed at point M_{1} close to the triplet resonance point, where the relaxation of T_{23} is suppressed by the exchange coupling. The final charge configuration is probed by integrating the sensor signal for 4 μs and the outcome is mapped to either the singlet or one of the triplets^{13,33}. To cancel out the slow drift of the background signal, the charge sensor signal in (102) is recorded before every spin manipulation at point O and it is subtracted from the integrated signal at M_{1}. The wholepulse cycle is repeated for N = 200 times to infer the singletreturn probability^{13,33} P_{S}.
During the acquisition of the data in Figs. 3 and 4a, the spectrum of ΔB_{ ij } is broadened by the Overhauser field fluctuation^{5,21}. To make the fluctuation spectrum homogeneous for the wholedata set in a single acquisition, independent variables (t_{evolve} and ε) are scanned consecutively to obtain single samples for each point and the whole scan is repeated N = 200 times. This is essential to distinguish oscillation frequencies obtained at different values of ε without being disturbed by the fluctuation of ΔB_{ ij }. The singletreturn probability P_{S} is inferred from the ensemble average for each set of independent variables. The wholepulse cycle is finished in 22.7 s, which is shorter than the typical nuclear spin decorrelation time. This allows us to obtain a moderate inhomogeneous dephasing time of \(T_2^ \ast \approx 60{\kern 1pt} {\mathrm{ns}}\), though it also brings about a drift in ΔB_{ ij } from measurement to measurement^{5}.
The dephasing measurement in Fig. 5 is performed in a similar pulse cycle but with the twospin ‘SWAP’ pulse sequence shown in Supplementary Fig. 1b. After the system is initialized at M_{0}, the detuning is displaced rapidly to cross the S_{3}–T_{+23} resonance line, and then ramped slowly (within 1 μs) to ε = 0. During this slow passage, \(\left {\sigma _1{\mathrm{S}}_3} \right\rangle\) is loaded into the eigenstate of the local Zeeman field, \(\left {\sigma _1 \downarrow _2 \uparrow _3} \right\rangle\). The detuning is then positively (negatively) displaced to turn the exchange coupling J_{23} (J_{12}) on. During the hold time of τ_{SWAP}, spins in QD2 and QD3 (QD1 and QD2) are swapped at frequency \(J_{ij}\left( \varepsilon \right)/h\). Readout is performed after the detuning is pulsed back in reverse steps, where \(\left {\sigma _1 \downarrow _2 \uparrow _3} \right\rangle\) returns to \(\left {\sigma _1{\mathrm{S}}_3} \right\rangle\) in the (102) configuration while \(\left {\sigma _1 \uparrow _2 \downarrow _3} \right\rangle\), \(\left {\sigma _1 \uparrow _2 \uparrow _3} \right\rangle\), and \(\left {\sigma _1 \downarrow _2 \downarrow _3} \right\rangle\) are unloaded to triplet components, \(\left {\sigma _1{\mathrm{T}}_{23}} \right\rangle\) and \(\left {\sigma _1{\mathrm{T}}_{ \pm 23}} \right\rangle\), staying in (111).
We also carried out the measurement of the electron spin resonance (ESR) signals and the Rabi oscillations to verify the energy spectroscopy and characterize the performance of QD13 as spin qubits. The results are shown in Supplementary Fig. 2, which are taken with the pulse sequence in Supplementary Fig. 1c. The spin states are initialized at point I near the (101)–(102) charge transition line in Fig. 2c, where an electron in an excited state can escape to a reservoir and the doubly occupied singlet state S_{3} is formed. The state \(\left {\sigma _1{\mathrm{S}}_3} \right\rangle\) is then loaded into \(\left {\sigma _1 \downarrow _2 \uparrow _3} \right\rangle\) by the slow adiabatic passage. To drive the electron spin resonance with the micromagnet proximity field^{8,9}, we applied a microwave burst of duration t_{burst} = 1 μs to the gate electrode shared by the three QDs (horizontal fine gate in Fig. 2b). This leads to the mixing of \(\left {\sigma _1 \downarrow _2 \uparrow _3} \right\rangle\) with \(\left {\sigma _1 \uparrow _2 \uparrow _3} \right\rangle\) (\(\left {\sigma _1 \downarrow _2 \downarrow _3} \right\rangle\)) when QD2 (QD3) is in resonance. The DC voltage applied to the shared gate was negatively offset during the microwave burst to prevent leakage of electrons to the reservoirs. Finally, readout is performed at point M_{0}, where the relaxation time of two triplet states, \(\left {{\mathrm{T}}_{ + 23}} \right\rangle = \left { \uparrow _2 \uparrow _3} \right\rangle\) and \(\left {{\mathrm{T}}_{  23}} \right\rangle = \left { \downarrow _2 \downarrow _3} \right\rangle\), is sufficiently long. The two lines are separated by ≈8 mT with the effective gfactor of \(\left g \right = 0.34\), in agreement with \(\Delta B_{23}/h = 45\,{\mathrm{MHz}}\) found in Fig. 4. Similarly, the ESR lines for QD1 and QD2 shown in the upper panel were taken by preparing \(\left { \downarrow _1 \uparrow _2\sigma _3} \right\rangle\) adiabatically loaded from \(\left {{\mathrm{S}}_1\sigma _3} \right\rangle\). The Zeeman field difference is found to be ≈18 mT, which is reasonably close to the value of \(\Delta B_{12}/h = 45\,{\mathrm{MHz}}\) used in the main text within the magnitude of the Overhauser field fluctuation. The Rabi oscillations of QD13 were taken by repeating the singleshot measurement cycle with t_{burst} increased consecutively from 30 ns to 1.8 μs in 60 steps (Supplementary Fig. 2b). We then calculate the sliding Gaussian average of P_{S}, first against the microwave frequency with the standard deviation of σ_{ f } = 0.2 MHz, and then against t_{burst} with σ_{ t } = 30 ns.
Model of the coherent dynamics
We describe our spin chain in the FermiHubbard model and the Hamiltonian of the relevant energy levels reads
where S_{ i } denotes the doubly occupied singlet state in QDi and \(\hat{s}_i^z\) is the z component of the spin operator. The local Zeeman energies are given by \(B_1 = \left g \right\mu _{\mathrm{B}}B_{{\mathrm{ext}}}  \Delta B_{12}\), \(B_2 = \left g \right\mu _{\mathrm{B}}B_{{\mathrm{ext}}}\), and \(B_3 = \left g \right\mu _{\mathrm{B}}B_{{\mathrm{ext}}} + \Delta B_{23}\) (note that B_{ i } and ΔB_{ ij } are defined in units of energy) with g the electronic gfactor and μ_{B} the Bohr magneton. To explain the experimental results, we find it necessary to take into account the detuningdependent interdot tunnel couplings. Here we adopted phenomenological functions of the forms \(t_{12} = t_{\mathrm{L}}{\rm exp}\left[ {  \left( {\frac{{\varepsilon  \varepsilon _{\mathrm{L}}}}{w}} \right)^2} \right]\) and \(t_{23} = t_{\mathrm{R}}{\rm exp}\left[ {  \left( {\frac{{\varepsilon  \varepsilon _{\mathrm{R}}}}{w}} \right)^2} \right]\) with w/h = 30.0 GHz, which are similar to those used in the previous TQD experiments^{2,15,34}.
The Hamiltonian in Eq. 2 is diagonalized using the program^{35} QuTiP. To fit the experimental data in Fig. 4b with calculated energy gaps, we also take into account the S_{1}–T_{+12} and S_{3}–T_{+23} resonance points which appear as faint vertical lines at ε_{L}–ε = B_{1} + B_{2} and ε–ε_{R} = B_{2} + B_{3} for t_{evolve}>100 ns. These constraints together with the spectral lines in Fig. 4b are sufficient to determine all the unknown parameters in Eq. 2. We also confirmed that the derived energy scale of the detuning axis agrees with the photonassisted tunneling signal. The calculated eigenenergies for the derived parameters are shown in Fig. 2d and Supplementary Fig. 3.
The coherent spin dynamics of the system is calculated by numerically solving the Lindblad master equation,
where \({\cal L}_n\) are Lindblad operators describing the coupling between the system and the environment. The detuning ε is varied as a function of time to simulate the approximate pulse waveforms used in the experiment. Among several possible decoherence sources, the spin relaxation is unimportant because the relaxation time is much longer than the submicrosecond timescale of interest. Instead, the fluctuation of the energy levels leads to pure dephasing in the spin dynamics. The energy levels E_{ k } as described by Eq. 2 are susceptible to the charge noise in the electrostatic potentials defining ε, ε_{L,R}, and t_{L,R} as described by Eq. 1. The choice of the ratio \(\gamma _{\varepsilon _{{\mathrm{L,R}}}}/\gamma _{t_{{\mathrm{L,R}}}} = 100\) is somewhat arbitrary for the fits in Fig. 5, and it does not affect the conclusion. Here we used \(\gamma _{\varepsilon _{{\mathrm{L,R}}}}/\gamma _{t_{{\mathrm{L,R}}}} \gg 1\) for consistency with previous studies^{6}. The dephasingassisted adiabaticity of the spin swap process is clearly visible in the simulation result shown in Supplementary Fig. 4, where the population of \(\left { \downarrow _1 \uparrow _2 \uparrow _3} \right\rangle\) increases monotonically as the dephasing rate increases. We note that the population of \(\left { \uparrow _1 \downarrow _2 \uparrow _3} \right\rangle\) at ε = 0 also increases with the dephasing rate due to the increased adiabaticity across the transition at ε = ε_{R}. This suggests that the dephasing noise could also influence the visibility of the standard S–T oscillations observed in double QDs. However, the consequence may be subtler than the TQD case as the effect does not generate peculiar coherent states.
The charge noise in our system most probably originates from the Johnson–Nyquist voltage noise of the room temperature electronics. The noise is fed to gate electrodes through the broadband coaxial lines which attenuate the noise power at 10 GHz only by 12 dB (including the effect of the attenuators and the transmission loss) as measured by the vector network analyzer. The broadband spectrum of this noise contributes the most to the dephasing effect discussed in the main text, while the nuclear Overhauser field is quasistatic, having negligible impact on the coherent dynamics. The lowfrequency part of the voltage noise (below the 10 GHz bandwidth) adds up only to \(\delta \varepsilon _{{\mathrm{rms}}} \approx 0.1\,\upmu {\mathrm{eV}}\) in the rootmeansquare fluctuation of the detuning energy, which is well within the noise level commonly observed in similar devices^{36}. Although the fluctuations in ε, ε_{L,R}, and t_{L,R} could be correlated with each other due to finite crosstalk, this effect has no significant influence on the analysis.
While the charge noise in our system has no memory effect in the timescale of the spin dynamics (Markovian), the magnetic noise due to the Overhauser field fluctuation is much slower and mainly leads to the ensemble phase averaging effect. The magnetic noise overwhelms the charge noise only when Γ_{ kl } is reduced for \(\varepsilon _{\mathrm{L}} \ll \varepsilon \ll \varepsilon _{\mathrm{R}}\). We take this effect into account separately after the numerical calculation, by making the calculated oscillations decay with a Gaussian envelope of \(T_2^ \ast = 60\,{\mathrm{ns}}\). Similarly, the slow fluctuations of ε, which could arise from 1/flike noise, is taken into account by calculating the convolution with a Gaussian kernel of the standard deviation σ_{ ε } = 0.9 GHz.
The simulation result and the experimental data in Fig. 3 are in reasonable agreement for distant entanglement (red line and circles) without assuming an error in the singlettriplet readout. This reflects the high fidelity (≈90%) of the singleshot readout optimized for the measurement of this regime. On the other hand, we assume a lower readout fidelity of 75% for the triplet outcome in the case of nearestneighbor entanglement (blue line and circles). The change of this readout fidelity is also found as a change of the visibility near ε = 0 in Fig. 4a. This is probably due to the shift of the measurement point M_{1} caused by the transient effect of the detuning pulse.
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The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank S. Miyashita, S. Bartlett, and P. Stano for fruitful discussions. We thank RIKEN CEMS Emergent Matter Science Research Support Team and Microwave Research Group at Caltech for technical assistance. Part of this work was financially supported by the CREST, JST (JPMJCR15N2, JPMJCR1675), the ImPACT Program of Council for Science, Technology, and Innovation (Cabinet Office, Government of Japan), JSPS KAKENHI Grants No. 26220710 and No. 18H01819, and RIKEN Incentive Research Projects. T.O. acknowledges support from JSPS KAKENHI Grants No. 16H00817 and No. 17H05187, PRESTO (JPMJPR16N3), JST, Advanced Technology Institute Research Grant, the Murata Science Foundation Research Grant, Izumi Science and Technology Foundation Research Grant, TEPCO Memorial Foundation Research Grant, The Thermal and Electric Energy Technology Foundation Research Grant, The Telecommunications Advancement Foundation Research Grant, Futaba Electronics Memorial Foundation Research Grant, MST Foundation Research Grant, and Kato Foundation for Promotion of Science Research Grant. A.D.W. and A.L. greatfully acknowledge support from Mercur Pr20130001, BMBF Q.ComH 16KIS0109, TRR160, and DFH/UFA CDFA0506. X.H. thanks support by the US ARO. F.N. is partially supported by the MURI Center for Dynamic MagnetoOptics via the AFOSR Award No. FA95501410040, the JSPS (KAKENHI), the IMPACT program of JST, CREST Grant No. JPMJCR1676, RIKENAIST Challenge Research Fund, JSPSRFBR Grant No. 175250023, and the Sir John Templeton Foundation.
Author information
Affiliations
RIKEN Center for Emergent Matter Science, Wako, Saitama, 3510198, Japan
 Takashi Nakajima
 , Matthieu R. Delbecq
 , Tomohiro Otsuka
 , Shinichi Amaha
 , Jun Yoneda
 , Akito Noiri
 , Kenta Takeda
 , Giles Allison
 , Xuedong Hu
 , Franco Nori
 & Seigo Tarucha
JST, PRESTO, 418 Honcho, Kawaguchi, Saitama, 3320012, Japan
 Tomohiro Otsuka
Research Institute of Electrical Communication, Tohoku University, 211 Katahira, Aobaku, Aobaku, Sendai, 9808577, Japan
 Tomohiro Otsuka
Department of Applied Physics, University of Tokyo, 731 Hongo, Bunkyoku, Tokyo, 1138656, Japan
 Akito Noiri
 & Seigo Tarucha
Lehrstuhl für Angewandte Festkörperphysik, RuhrUniversität Bochum, D44780, Bochum, Germany
 Arne Ludwig
 & Andreas D. Wieck
Department of Physics, University at Buffalo, SUNY, Buffalo, NY, 142601500, USA
 Xuedong Hu
Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako, Saitama, 3510198, Japan
 Franco Nori
Physics Department, University of Michigan, Ann Arbor, MI, 48109, USA
 Franco Nori
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Contributions
T.N. and S.T. planned the project. A.L. and A.D.W. grew the heterostructure and T.N. fabricated the device. T.N. conducted the experiment with the assistance of M.R.D, T.O., J.Y., and A.N.; T.N. analyzed the data and performed numerical calculations with the help of X.H., M.R.D., S.A., J.Y., T.O., and F.N. All authors discussed the results and commented on the manuscript. The project was supervised by S.T.
Competing interests
The authors declare no competing interests.
Corresponding authors
Correspondence to Takashi Nakajima or Seigo Tarucha.
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