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A Heisenberg Spin Teleport

Quantum information science has the potential to revolutionize modern technology by providing resource-efficient approaches to computing. Quantum communication explores how superposition and entanglement can protect and transmit quantum information. Quantum teleportation, in which the quantum state of one system is transmitted to another, is a fundamental example of quantum communication. Here, we demonstrate a new way to transmit quantum spin states based on the Heisenberg interaction between electrons in solids. We teleport the spin state of an electron back and forth along a chain of electrons in a quantum-dot array without moving any electrons. We also demonstrate that this Heisenberg spin teleport is compatible with entangled states. This spin teleport may be especially useful for spin-based quantum information processing, where the transmission of quantum spin states across large quantum-dot arrays will be beneficial for universal quantum computation. This Heisenberg spin teleport also provides a striking example of the strange and interesting phenomena enabled by the laws of quantum physics.

Electron spin qubits most naturally interact with each other via direct wavefunction overlap, which generates Heisenberg exchange coupling [11]. In large-scale arrays of spin qubits, however, maintaining sufficient connectivity [23] for efficient and universal quantum computing poses a challenge. To this end, long-distance coupling between spins is an active area of research. Exciting possibilities include coupling spins to superconducting microwave photons [24][25][26], shuttling electrons between quantum dots via tunneling [27][28][29][30][31][32] or surface acoustic waves [33,34], and superexchange methods [35,36]. Theoretical proposals have also explored the possibility of a spin bus, in which always-on interactions between spins in a chain can transport spin states [37,38].
In this work, we present a new approach to the transmission of quantum spin states. We teleport the spin state of an electron back and forth across a quadruple quantum dot array, without ever moving any electrons. This method relies entirely on the quantum-mechanical exchange coupling of electrons in neighboring quantum dots. We also teleport one spin of an entangled pair to a distant electron and back. This "Heisenberg spin teleport" does not involve the motion of electrons, and it is compatible with arbitrary single-and multi-qubit states. The advantages of this scheme are that it does not require any additional quantum objects beyond the spins themselves, and it does not require any empty intermediate quantum dots, as electron shuttling often does. We expect that this spin teleport will be useful in the generation of long-range entangled states in spin-based quantum information processors. It also provides a vivid example of the unusual and useful phenomena made possible by the laws of quantum mechanics.
Our device is a quadruple quantum dot, which we fabricated in a GaAs/AlGaAs heterostructure [Figs. 1(a)-(b)]. Unlike previous GaAs quantum dot designs, we used a fully-overlapping device architecture [20,39,40] involving four layers of aluminum gates deposited on top of each other. This device architecture enables tight control of the electronic confinement potential, which is essential for the results shown here. We tune the device to the single occupancy regime, in which each dot is occupied by a single electron. The tune-up process is greatly facilitated by the use of "virtual gates" [27,31], which enables independent adjustment of the chemical potentials of the quantum dots [ Fig. 1(c)]. In our approach, we correct for the capacitive coupling of all barrier and plunger gates to the chemical potential of each dot. Thus, adjusting the chemical potential of each dot involves changes to all four plunger gates, and changing the tunnel barrier between a pair of dots involves changes to that barrier gate and all four plunger gates.
We initialize and measure the array by configuring it as a pair of singlet-triplet qubits (see Methods) [41,42]. (a) Scanning electron micrograph of the quadruple quantum dot. The lateral spacing between dots is 200 nm. All plunger gates (red) and inner barrier gates (blue) are connected to separate arbitrary waveform channels for independent pulsing. Ohmic contacts to the two-dimensional electron gas are indicated by an "x" inside of a square. The two quantum dots above the middle green gate are charge sensors, and their ohmic contacts are configured for rf-reflectometry. A grounded top gate (not shown) covers the active area of the device. (b) Line cut through the device at the position indicated in panel (a) showing the locations of all electrons. (c) Schematic potential landscape imposed by the confinement gates. In general, the plunger gates control the chemical potential µi of each dot i, and the barrier gates control the tunnel coupling between dots i and j, Tij . Ga and As nuclear spins (yellow and purple) contribute a random magnetic field at the site of each dot via the hyperfine interaction.
Each singlet-triplet qubit occupies a pair of quantum dots. In the following, we will refer to the left and right sides of the quadruple dot. We can initialize either side as | ↑↑ , | ↓↑ , |S = 1 The orientation of the spins in the |↓↑ state depends on the local magnetic gradient. In this work, magnetic gradients result from the hyperfine interaction between the electron and nuclear spins [43]. We use nuclear pumping and feedback to stabilize the gradient on the right side [42,44], and we empirically observe that the gradient is metastable on the left side, 20 40 60 Time (ns) Exchange oscillations between spins in dots 3 and 4. The detuning of the right side ǫR = µ3, with µ4 = −µ3. ǫR = 0 at the (0,2)-(1,1) crossing. (c) Exchange oscillations between spins in dots 2 and 3 measured by preparing the left side in the |T+ = | ↑↑ state and the right side by adiabatic separation of the singlet state. Oscillations were measured on the right side of the array. The detuning of the middle pair ǫM = 0 when ǫL = 10mV and ǫR = 6mV. Changes to the detuning of the middle pair are such that ∆ǫM = ∆µ2, with ∆µ3 = −∆µ2.
The visibility in this case is not as high as the other two panels, because we did not stabilize the magnetic gradient on the right side for this measurement. In all cases P as discussed further in Methods. We measure both sides of the array via spin-to-charge conversion through the Pauli spin blockade [41]. We readout in the {|S , |T } basis for each side of the array, where |T is any one of the triplet states {| ↑↑ , 1 √ 2 (| ↑↓ + | ↓↑ ) , | ↓↓ }. Adiabatic charge transfer of the electrons on each side into dots 1 and 4 maps | ↓↑ → |S , and all other product states to triplets. Diabatic charge transfer from the outer dots preserves the spin states, and diabatic transfer into the outer dots during readout projects a joint spin state onto the {|S , |T } basis [41].
We induce exchange coupling between two electrons by applying a voltage pulse to the barrier gate between them [45,46]. The voltage pulse creates an overlap between the wavefunctions of neighboring electrons, and the spins evolve according to the Heisenberg exchange Hamiltonian H int = h J 4 (σ x ⊗ σ x + σ y ⊗ σ y + σ z ⊗ σ z ). Here J is the coupling strength, σ x , σ y , and σ z are Pauli matrices describing the spin components of each electron, Step number 0 2 4 6 Step number Step number Step number (d) Four-dot teleport data. The four-dot teleport is described by the sequence S23, S12, S12, S23, S34, S34. In panels (a)-(d), the upper graph shows measurements on the right side, and the lower graph shows measurements on the left side. In each graph, the inset gives the expected outcomes. (e)   Barrier-induced exchange coupling between fully separated electrons is first-order insensitive to charge noise [45,46] associated with the plunger gates, which would otherwise randomly shift the locations of the electronic wavefunctions and promote rapid decoherence. This insensitivity to noise is critical for the results we describe below. Figure 2 demonstrates coherent exchange oscillations between all nearest-neighbor pairs of spins in the array.
To teleport the spin state of an electron, we initial-ize the array in the | ↑↑↓↑ state, and we concatenate different SWAP operations between pairs of electrons.
To swap the spins of the electrons in dots 3 and 4, for example, we apply a voltage pulse to barrier gate T 34 , timed to give a π pulse. We denote this operation as S 34 . In general, we use S ij to mean a SWAP operation between spins i and j. During each barrier pulse, the chemical potentials of the dots themselves remain fixed, because we compensate for gate crosstalk, and the electrons do not move. Typically, exchange pulses are < 10ns in length, and are usually 3π pulses to ensure that exchange strengths are larger than magnetic gradients, as discussed below.
We begin by teleporting the down spin originally associated with the electron in dot 3 through the following sequence of operations: S 34 , S 34 , S 23 , S 23 , S 34 , S 34 . Before the sequence begins and after each step, we measure both sides of the array (in the {|S , |T } basis) to confirm the expected spin states [ Fig. 3 ΔB24   S23  ΔB24  S23   I,ΔB,I  S23,ΔB,I  S23,ΔB,S23  I,ΔB,I  S23,ΔB,I  S23, each successive step. We discuss the limiting factors in this teleport process below. We emphasize that the electrons themselves do not move during this process. It is only the spin-down state, which was originally associated with the electron in dot 3, that moves. The teleportation of the spin state in this way is a direct consequence of the Heisenberg exchange interaction, and it is closely related to the entanglement between the electrons that occurs during this process. The imperfect visibility of the prepared triplet states on the left side is due to thermal population of excited spin states, as discussed in the Methods section.
As a check, we have verified that eliminating certain pulses in the spin teleport sequence also produces the expected results. Figure 3(b) displays the outcome when we replace S 34 with the identity operation I, implemented as a wait with no barrier pulse. The data show the expected result. Likewise, we have checked that replacing S 23 with I also gives the expected result in Fig. 3(c).
We also demonstrate that we can teleport a spin back and forth across the full four-dot array. We apply the following swap sequence: S 23 , S 12 , S 12 , S 23 , S 34 , S 34 . For this sequence, we expect the following measurements on the right side: S, T, T, T, S, T, S and the following on the left: T, T, S, T, T, T, T . The expected trend is clearly evident in the data [ Fig. 3(d)].
Having demonstrated the feasibility of teleporting single-spin eigenstates, we now demonstrate teleportation of entangled states (Fig. 4). By electronic exchange with the reservoirs and diabatic charge transfer, we prepare the array in the 1 √ 2 | ↑↑ ⊗(| ↑↓ − | ↓↑ ) state. Then, we apply an S 23 operation [ Fig. 4(a)]. This operation causes the singlet state, which was initially prepared in dots 3 and 4, to reside in dots 2 and 4.
In general, a separated singlet state in dots i and j will evolve to the unpolarized triplet state |T 0 and back, if there exists a magnetic-field difference ∆B ij between quantum dots i and j [42,43,47]. The singlet-triplet oscillation frequency is gµ B ∆B ij /h, where g is the electron g factor, µ B is the Bohr magneton, and h is Planck's constant. In our experiment, after the separated singlet state evolves for a variable period of time around the magnetic gradient, we apply an S 23 operation, bringing the singlet back into dots 3 and 4. We then measure the right side of the device in the usual {|S , |T } basis after diabatic charge transfer. Provided the S 23 operations preserve the entangled state, we expect to observe coherent singlet-triplet oscillations, corresponding to evolution around ∆B 24 .
We observe clearly-visible singlet-triplet oscillations [Figs. 4(b)-(c)]. Simulations conducted by integrating the Schrödinger equation for a three-spin system show excellent agreement with the data [Fig. 4(d)], confirming successful teleportation of an entangled pair of electrons.
We note that such a ∆B measurement across two dots is a routine procedure for singlet-triplet qubits when the two electrons are separated to neighboring dots via tunneling [41,44,47]. Recently, ∆B oscillations between singlet pairs separated to distant dots via tunneling have also been observed [28,29]. In the present work, however, the electrons themselves do not move. We use purely quantum-mechanical means to change the physical location of the entangled spin state.
As a check, we have performed the same experiment while omitting both SWAP operations. In this case, we also observe oscillations around the magnetic gradient, but with a different characteristic frequency, corresponding to ∆B 34 Figure 4(e) shows the time evolution of ∆B 24 and ∆B 34 during the course of the experiment. Since both field gradients result from different random nuclear spin ensembles, we expect their time evolution to be different, as we observe [ Fig. 4(e)].
When we omit only the final SWAP operation, we observe small-amplitude oscillations, consistent with our simulations [Figs. 4(c)-(d)]. A perfect initial S 23 operation would completely transfer the entanglement between dots 3 and 4 to dots 2 and 4, and we would not expect to observe oscillations without a final S 23 . However, our S 23 operation is imperfect, because the magnetic gradient ∆B 23 prevents a pure exchange rotation. After this imperfect SWAP operation, the electron spin in dot 3 remains weakly entangled with the electron spin in dot 4, and weak ∆B 34 oscillations are observed. As expected, the oscillation frequency in this case clearly corresponds with ∆B 34 .
The evidence confirming that we have teleported an entangled state is that we observe ∆B oscillations with a different frequency if we apply S 23 operations before and after an evolution period, compared to the case when we do not apply these operations. To ensure that differences between these cases do not result from a randomly changing nuclear magnetic field between experiments, but instead result from the teleportation of entangled states, we have interleaved the averaging of these measurements in time as discussed further in the Methods section.
We can also coherently teleport one spin state of an entangled singlet pair to the other end of the array and back. We achieve this by applying S 23 and S 12 operations before and after free evolution around the magnetic gradient. The data for this experiment are shown in Extended Data Figure 4.
The primary limiting factors of the teleport operation are the presence of a magnetic gradient between the dots and the temporal fluctuations in this gradient, result-ing from the nuclear spin noise. In general, exchange coupling tends to swap the spin state of two spins, but a magnetic gradient ∆B tends to drive transitions to the singlet or unpolarized triplet configurations of the two spins [41,43]. The presence of a magnetic gradient therefore makes single-pulse pure exchange rotations impossible. In this work, we minimized this effect by using exchange strengths of several hundred MHz. Typical gradient strengths were several tens of MHz. Because we do not perform pure exchange rotations, the final joint spin state of a pair of spins after a SWAP operation is also not an eigenstate of the local magnetic gradient. Thus, after a typical SWAP operation in our experiment, the magnetic gradient causes the joint spin state of two electrons to continue to evolve, further diminishing the fidelity of the teleport operation. A final complication for the present device is that the nuclear magnetic field fluctuates in time as a result of the changing nuclear spin polarization. This spin noise creates second-order noise in the exchange splitting, which increases dephasing. We included these three effects in numerical simulations of our spin teleport, and we find good agreement with our data, as shown in Extended Data Figs. 2 and 3.
In general, we expect the spin teleport operation to work best with small magnetic gradients. However, it is also likely that dynamically corrected exchange gates could implement high-fidelity exchange rotations in the presence of gradients [48]. We also expect that the spin teleport can work even better in silicon qubits, where nuclear fluctuations are suppressed.
We have demonstrated a Heisenberg spin teleport by transmitting the spin state of an electron back and forth along an array of electrons in a quadruple quantum dot. We have shown that we can teleport both single-spin eigenstates and entangled states. It is interesting to compare this Heisenberg spin teleport to quantum teleportation. Conventional quantum teleportation relies on an existing entangled pair of qubits that are separated to different locations. Here, we do not begin with an entangled state of two qubits. However, during the swap process, the two spins evolve through an intermediate entangled state before arriving in the swapped sate. Thus, quantum entanglement remains an essential feature for the transmission of qubit states. Although the method we use in this work to transmit spin states differs from quantum teleportation, the end result is the same: a quantum state has vanished from one qubit and appeared on another. In the future, we expect that this Heisenberg spin teleport will be useful in spin-based quantum computing for the transmission of spin states along large arrays and also for the generation of long-range entangled states. More broadly, our work illustrates in a new way how it is possible to transmit the quantum state of an object without moving the object itself. As such, the Heisenberg spin teleport provides a vivid example of the exciting and intriguing potential of quantum physics for the transmis-

Device
The quadruple quantum dot is fabricated on a GaAs/AlGaAs hetereostructure with a two-dimensional electron gas located 91 nm below the surface. The twodimensional electron gas density n = 1.5 × 10 11 cm −2 and mobility µ = 2.5 × 10 6 cm 2 /Vs were measured at T = 4K. Voltages applied to Al depletion gates define the quadruple-dot potential. In addition to the three layers of aluminum gates shown in Fig. 1, we also found it necessary to deposit a grounded top gate over the device, likely to screen the effects of disorder in the twodimensional electron gas, perhaps imposed by the 10nm-thick aluminum oxide layer we deposited via atomic layer deposition. The quadruple dot is cooled in a dilution refrigerator to a base temperature of approximately 10 mK. An external magnetic field B=0.5 T is applied in the plane of the semiconductor surface perpendicular to the axis connecting the quantum dots. This orientation of the magnetic field ensures effective dynamic nuclear polarization [49].

Initialization and readout
To initialize the array, we configure it as a pair of singlet-triplet qubits [41,42]. We load two electrons in the singlet configuration in dots 1 and 4 each via electron exchange with the reservoirs [42]. If we diabatically separate the electrons, they remain in the singlet state. We can also adiabatically separate the electrons into neighboring dots such that each dot has one electron. Upon adiabatic separation, the singlet states evolve into product states, with one electron spin-up and the other spindown in each pair. The orientation of the spins is determined by the local magnetic field gradient. In the present case, the magnetic gradient results from the hyperfine interaction between the electron and Ga and As nuclear spins, each of which have nuclear spin 3/2 [43]. For this work, we empirically observe that the magnetic gradient of dots 1-2 is metastable, and the gradient usually favors spin down in dot 1 and spin up in dot 2. We use dynamic nuclear polarization and feedback [42,44,47,49] to set the magnetic gradient of dots 3-4, such that the ground state is spin down in dot 3 and spin up in dot 4. We have used a sequence of exchange oscillation measurements to verify that the ground state of the quadrupole dot array, initialized in this way, is | ↓↑↓↑ , as we expect (Extended Data Fig. 1). It is also possible to initialize either pair of quantum dots in the |T + = | ↑↑ configuration by electron exchange with reservoirs [42] when the ground state of a pair of electrons has one electron in each dot.
The assumption of metastability of the left-side gradient does not affect the data. It only affects our prediction for the measurement outcomes. If this assumption were violated at any time, it would appear to diminish the apparent agreement between our data and our predictions.
The prepared triplet states in Fig. 3 do not appear with perfect visibility. This reduction in visibility occurs because the Zeeman energy of the electrons at B = 0.5T is gµ B B/k B ≈ 150 mK, which is not significantly higher than the thermal energy, and excited spin states remain populated to a small degree. Increasing the magnetic field or decreasing the temperature could improve the triplet visibility.
After manipulating the spins, we read them out by adiabatically moving the electrons in dots 1 and 2 both into dot 1, and we move the electrons in dots 3 and 4 into dot 4. If the joint spin state of each pair evolves into the singlet state during adiabatic transfer, both electrons can occupy the same dot [41]. However, if the pair evolves to a triplet state (if they have the same spin, for example), the Pauli exclusion principle forbids both electrons from occupying the ground state of the outer dot, and the pair remains separated. We detect this change in charge configuration through rf-reflectometry of proximal sensor quantum dots [50]. In addition to conventional Pauli spin-blockade, we also use a shelving mechanism to increase the visibility of the readout [51], and we can achieve single-shot readout within 5µs integration times.
All data presented here were taken by reading out the two sides of the array sequentially. Specifically, for each single shot measurement, we read out only one side. Although we applied exactly the same initialization and exchange pulse sequence when reading out different sides, we apply a different readout sequence depending on the side, as we discuss below. Sequential readout of the sides is sufficient to demonstrate teleportation of single-spin eigenstates, because single-shot correlations are not required.
We observe that reading out both sides of the array during the same single-shot measurement results in significant state-dependent crosstalk on the left-side signal from the right side. This effect results from the capacitance between the right and left sides of the array. Although the idling configuration of each side is in the (0,2) charge configuration, exchange pulses cause each side sometimes to occupy the (1,1) charge configuration due to the Pauli spin blockade, and we do not know ahead of time which charge configuration each side will have for a given single-shot measurement. Changes in the charge configuration of one side shift the charge stability diagram of the other side, and these shifts interfere with the measurement process. In particular, we observe that when the right side is in the (1,1) charge configuration, the left side experiences rapid relaxation to the (0,2) singlet state during adiabatic transfer to the readout position. We believe this results from inadvertent electron exchange with the reservoirs on the left side when the right side occupies the (1,1) charge configuration.
We solve this problem by adiabatically transferring the left-side electrons only when the right side occupies the (0,2) charge configuration. Specifically, we adiabatically transfer from (0,2) to (1,1) on the left side before the right side, and we transfer back to (0,2) on the left side after the right side.
In addition, to readout the left side, we reload the right side as an (0,2) singlet after adiabatic transfer to the right-side readout position but before adiabatic transfer to the left-side readout position. This step ensures that the right side has the same charge configuration every time the electrons on the left side separate and recombine. We verified that the presence or absence of this reload on the right side has no discernible effect on leftside exchange measurements, and it removes the statedependent crosstalk effect. To readout the right side, we omit the extra reload and enforce a wait for the same length of time. We use this protocol to take the data for all panels in the paper. We emphasize that the extra initialization step on the right side always took place after all exchange pulses were finished, and exactly the same initialization and exchange pulses were applied in the sequences to readout both the right and left sides of the array.
We have also observed similar crosstalk effects from the left side on the right side. In general, we observe that crosstalk effects depend sensitively on device tuning and may also partly result from an imperfect gate capacitance matrix. However, for the tuning used for the experiments described here, left-to-right crosstalk was not significant.
To demonstrate teleportation of entangled states, we only measured the right side of the array. Because we measure the right side directly in the singlet-triplet basis, measurement of a single side is sufficient to distinguish evolution between these entangled states.

Interleaved measurements
We interleaved the averaging of different pulse sequences to demonstrate teleporation of entangled states. The purpose of interleaving the measurements was to ensure that changing nuclear fields did not confound the measurement, since we rely on observing coherent oscillations of different frequencies. Specifically, we performed 32 single shot experiments (initialization, evolution, and measurement), each lasting 28 µs, omitting both S 23 operations. Immediately following this set, we performed 32 single-shot measurements omitting only the second S 23 operation, and then we performed 32 singleshot measurements with both S 23 operations. We then averaged each set of 96 measurements 512 times, and the averaged result is displayed as one line in Fig. 4(b). Each line takes approximately 1 second. Empirically, we find nuclear magnetic fields are reasonably stable on this timescale [47]. We repeated this process 16 times. As can be seen in Fig. 4(b), each line shows coherent ∆B oscillations. It is also evident that averaging all lines together would show significant dephasing. Note that in this experiment, we stabilized ∆B 34 using nuclear pumping.

Simulation
To generate the simulation in Fig. 4(d), we numerically integrated the Schrödinger equation for a three spin system. We generated a simulated SWAP operation from the following Hamiltonian: We assumed a fixed exchange coupling of J 23 of 150 MHz between spins 2 and 3, and we adjusted the time for the SWAP operation to give a 3π pulse. These parameters correspond closely to the actual experiments. We adjusted the local nuclear magnetic fields B k of spin k to be (0, 75, 35) MHz × 2h gµB in dots 2-4. These were adjusted to match the frequencies observed in Fig. 4(c).
We initialized the three-spin system in the | ↑ ⊗ 1 √ 2 (| ↑↓ − | ↓↑ ) state, corresponding to dots 2-4. After applying an S 23 operation (including the effects of magnetic fields), we evolved the system for a variable evolution time in the presence of the magnetic fields. Then we applied a final S 23 operation, and we projected the resulting state along all states with a singlet in dots 3 and 4. To generate the simulated control measurements, we omitted the relevant S 23 operations.
We have also simulated the single-spin teleport in a similar way. We numerically integrated the Schrödinger equation for a 4-spin system. We choose the nuclear magnetic fields for each site to be approximately (0, 50, 100, 150) MHz × 2h gµB , corresponding to a 50 MHz gradient between dots. We allow the gradient to fluctuate by 20 MHz on each dot between runs. We assume J = 200 MHz for each exchange pulse, and we set the pulse time to generate a 3π rotation. We also include a 3-10 ns wait between exchange pulses, which we used in the experiments. We also included thermal population of excited states during the T + loading process by assuming an electron temperature of 150 mK. We applied the pulse sequences described in the main text, and then we projected the left and right sides onto final states with the | ↓↑ configuration on either side, corresponding to the singlet outcome after adiabatic charge transfer. We  We load the left side in the | ↑↑ state and the right side by adiabatic separation of the singlet state, which gives either |↑↓ or |↓↑ , depending on the sign of the gradient. We pulse T23 to induce exchange between the middle two spins. Dynamic nuclear polarization with singlets yields no oscillations, while pumping with triplets yields oscillations. These data confirm that the separated singlet state evolves to the |↓↑ state under triplet pumping for the right side. (d) Proof of the ground state of the left side. We initialize the array by separating singlets on both sides. In the case of triplet pumping on the right side, the third spin is |↓ , so it must be the case that the second spin is |↑ in order to generate exchange oscillations with a T23 pulse, as measured on the left side. Singlet pumping on the left side yields no exchange oscillations. (e) The same initialization and pulses as (e), but measured on the right side. In all cases, P L(R) S indicates the singlet return probability measured on the left(right) side. Step number Step number Step number   Figure 3. Results of the four-dot teleport simulation, as described in the Methods section, showing good agreement with the data in Fig. 3. (a) Simulated right-side measurements for the S23, S12, S12, S23, S34, S34 sequence. (b) Simulated left-side measurements for the same sequence.