Probabilistic teleportation of a quantum dot spin qubit

Electron spin s in semiconductor quantum dot s have been intensively studied for implementing quantum computation and high fidelity single and two qubit operation s have recently been achieved . Quantum teleportation is a three qubit protocol exploiting quantum entanglement and it serv es as a n essential primitive for more sophisticated quantum algorithm s Here, we demonstrate a scheme for quantum teleportation based on direct Bell measurement for a single electron spin qubit in a triple quantum dot utilizing the Pauli exclusion principle to create and detect maximally entangled state s . T he single spin polarization is teleported from the input qubit to the output qubit with a fidelity of 0.9 1 We find this fidelity is primarily limited by singlet triplet mixing which can be improved by optimizing the device parameters Our results may be extended to quantum algorithms with a larger number of se miconductor spin qubit s


Introduction:
An electron spin qubit in semiconductor quantum dot 1 is a promising building block for quantum computing. Recent progress has realized fundamental control on single and two qubits [2][3][4][5] . Implementing three-qubit algorithms is a significant step forward, as they can allow for demonstrations of key primitive algorithms such as encoding a single logical qubit 6 . Quantum teleportation 7 (QT) is an attractive instance of three qubit algorithms and has been demonstrated in many physical systems [8][9][10][11] because it enables long range quantum communication via quantum repeaters 12 as well as computational models such as gate teleportation 13 and measurement-based control 6 . In quantum-dot spin qubits, however, QT 8,14 has been demonstrated only recently 15 , employing a SWAP operation in a Heisenberg spin chain to distribute quantum entanglement in a quadruple quantum dot.
Here, we design a probabilistic QT protocol, where an entangled state is distributed by direct transfer of the qubit with a simple linear ramp pulse in a semiconductor triple quantum dot (TQD) device, and demonstrate the teleportation of a single-electron spin qubit. We employ a sequence of input qubit state preparation, the QT protocol and output qubit state readout by energy-selective tunneling (Fig. 1a). We show that the spin polarization of the input qubit is teleported to that of the output qubit when and only when we have access to the outcome of the Bell measurement. This agrees with the essential property of QT requiring not only quantum entanglement but also classical information about an outcome of the Bell measurement. We analyze possible error sources in our QT process based on input-output relation and find that the teleportation infidelity originates primarily from the leakages of singlet states to and from spin polarized triplet states in the preparation and the measurement of the singlets.

Results:
The QT protocol is created in Q2 and Q3 as the maximally entangled pair by initializing a doubly-occupied spin singlet in QD3 and the electron hosting Q2 is moved to QD2. Then, the electron is attempted to be transferred to QD1. In this process, the Bell measurement of Q1 and Q2 is implemented by singlet-triplet readout using Pauli spin blockade (PSB) 16,17 , where the tunneling of the electron hosting Q2 into QD1 coincides with the projection of Q1 and Q2 onto the . The Bell measurement only distinguishes the singlet from the other three maximally entangled states, which makes success of the QT stochastic. The state of Q3 becomes the same as Q1 when the outcome of the Bell measurement is singlet. The key ingredients in our QT protocol are coherent separation of | 23 〉 and detection of | 12 〉. Our protocol realizes these steps not with two-qubit gates but via rapid adiabatic passage 16 employing a linear ramp of the detuning energy , where is defined as the energy difference between (2,0,1) and (1,0,2) (see Fig.1d). An advantage of this approach is that the pulse sequence is less complex and the entire operation time is within nanoseconds under appropriate inter-dot tunnel couplings.

Triple quantum dot device
A linearly coupled TQD 18,19 is fabricated on a GaAs/AlGaAs heterostructure wafer as shown in Fig. 1c. We apply voltage pulses to the P1 and the P3 gates to rapidly control energy levels of the TQD. A micromagnet fabricated on the wafer surface forms an inhomogeneous local magnetic field and enables addressable electric-dipole spin resonance (EDSR) 3,20 control. The local magnetic field is largest in QD1 followed in order by in QD2 and QD3 (the static external magnetic field is 3.07 T). We detect the electron charge configuration in the TQD by measuring the conductance of the nearby sensor dot as demodulated reflectometry signal ( rf ) 21 . Figure 1d shows the charge stability diagram of the TQD around the charge states of (N 1 ,N 2 ,N 3 ) = (1,1,1), (1,0,2) and (2,0,1) used in this work, where N i denotes the number of electrons in QDi. Note that we tune the inter-dot tunnel couplings so that direct spin-spin interaction between QD1 and QD3 in (1,1,1) is negligible (see Supplementary I).

Ingredients of the QT protocol
The fidelity of the entire process of our QT protocol is subject to the tunnel coupling strengths because our approach relies on the mapping between the singlet states and the doubly occupied charge states during detuning ramps. For example, the weak inter-dot tunnel couplings may cause failure in detection of | 12 〉 and separation of | 23 〉 due to slow electron tunneling of Q2 (see Fig. 1b). To confirm the feasibility of | 12 〉 detection, we perform the PSB measurement with QD1 and QD2 (Fig. 2a). Figure 2b shows the histogram of the single-shot PSB signal rf measured for QD1 and QD2 after loading a spin-up into QD1 and a random spin into QD2. We use a latched readout technique to enhance the readout visibility, which transfers the spin-blocked (1,1,1) charge state to (2,1,1) before the readout 22,23 . The solid line is a fit using two noise-broadened Gaussian distributions considering the relaxation of triplet states 17 . The state is registered as singlet (triplet) when rf is lower (higher) than the threshold voltage, threshold . Next, we measure singlet-triplet oscillation (ST oscillation) in QD2 and QD3 induced by the local Zeeman field difference 24 Δ to ensure the creation and separation of | 23 〉 (Fig. 2c). Figure 2d shows the measured singlet probability, , as a function of the dwell time dwell in (1,1,1).
Because the dwell point is far detuned from the (1,1,1)-(1,0,2) degeneracy point, the exchange coupling between Q2 and Q3 is suppressed and the observed periodic oscillation of indicates coherently repeated transitions between the singlet and non-polarized triplet 17 (The oscillation visibility is largely limited by readout error arising from the relaxation of nonpolarized triplet, which does not contribute to the teleportation infidelity). These results ( Fig. 2b and Fig. 2d) show that the device is properly set up to realize coherent separation of | 23 〉 and projection measurement onto | 12 〉.

Preparation of input qubit and readout of output qubit
To prepare an input state for the QT protocol, we rotate Q1 using resonantly driven coherent oscillation. We can use a micromagnet-mediated EDSR 20 for the qubit rotation, but here we find that we can manipulate Q1 with higher speed and less decay by resonant transitions between |↑ 1 ⇓〉 and |↓ 1 ⇑〉 in QD1 and QD2 (resonant SWAP) 25,26 than by EDSR. The double lines arrow (⇑ or ⇓) represents a spin that is temporarily loaded in QD2 to assist the rotation of Q1 and is later discarded to the reservoir. We implement this resonant SWAP by applying a microwave (MW) to the P2 gate after initializing a singlet state in QD1 and subsequently loading |↑ 1 ⇓〉 using slow adiabatic passage 27 (Fig. 2e). The resulting two-spin state is |↑ 1 ⇓〉 + |↓ 1 ⇑〉 with | | 2 + | | 2 = 1 . By emptying QD2, the coherence of Q1 is lost but the probabilities in the up/down basis | | 2 and | | 2 are retained. Therefore, | | 2 (| | 2 ) is equal to the spin-up (spin-down) probability of Q1, ↑,in ( ↓,in ). To estimate ↑,in , we measure the probability of |↑ 1 ⇓〉 ( ↑ raw ) after a MW burst. The measured probability distribution ↑ raw is influenced by the readout infidelities, 1 − ↑,in and 1 − ↓,in for the spin-up and -down state, respectively, and the discrepancy from the actual probability may lead to the underestimation of the performance of our QT protocol. We exclude the readout infidelities as ↑,in = ↑ raw + ↓,in −1 ↑,in + ↓,in −1 with ↑,in = 0.96 and ↓,in = 0.90 (see Supplementary II). Figure 2f shows ↑,in as a function of the MW burst time burst , indicating that we can vary the spin-up probability of the input state.
The output qubit state Q3 teleported from Q1 is read out by the spin-selective tunneling to the lower reservoir 28 . This readout is performed by pulsing gate voltages near the (2,0,1)-(2,0,0) transition line (marked by a star in Fig. 1d). The tunneling of Q3 to the reservoir occurs only when its spin is down. We estimate the output qubit readout fidelities for spinup and -down state to be ↑,out = 0.83 and ↓,out = 0.50 from additional experiments (see Supplementary III). ↓,out is limited by a relatively small tunnel rate to the reservoir compared to the readout time. We estimate the spin-up probability by taking into account those infidelitiessimilarly to Q1.

Demonstration of QT
We now integrate all operations including the preparation of Q1, our QT protocol and the final readout of Q3 in one sequence. Figure 3a shows the spin-up probability of Q3 obtained as a function of burst to drive Q1. The gray squares denote ↑,out , the spin-up probability produced without using the information of the Bell measurement. ↑,out is independent of burst , showing no correlation with the input spin Q1. In contrast, if we extract the data set conditioned on the singlet outcome in the Bell measurement (classical information, CI), we obtain ↑,out CI (see blue circles in Fig. 3a), which reproduces the Rabi oscillation of Q1 ( Fig. 2f) well. Here, threshold of the Bell measurement is chosen to take advantage of classical information maximally (see the black dashed line in Fig. 3B), i.e., to maximize the oscillation amplitude out of Q3 spin-up probability. As the accuracy of the classical information is degraded deliberately by raising threshold , a monotonic decrease of the amplitude is observed. These agree well with an essential property of the QT, that useful information cannot be extracted only from the local measurement of Q3 and the outcome of the Bell measurement is required to reproduce the original state of Q1. The difference between ↑,out and ↑,out CI and the similarity between ↑,in and ↑,out CI are the hallmark of successful teleportation.

Discussion
When there are no errors in the singlet preparation and detection, ↑,out CI obtained after our QT protocol should be identical to ↑,in . We plot these two probabilities against each other in Fig.4 (pink triangles). The discrepancy between the two suggests that such errors are indeed not negligible in our experiment. We discuss below the effects of those errors in our QT protocol.
The second source of error is in the detection process. This can be decomposed to the state-mapping error and the electrical detection error in the Bell measurement. The latched readout technique employed here helps to suppress the electrical detection error to 0.02 but a large state-mapping error may remain due to the nonideal inter-dot and dot-to-lead tunnel rates 22,31 . To model the error in the detection process, we use the measurement operator Bell as Bell = ,Bell | 12 〉〈 12 |+(1 − 0 ,Bell )| 0,12 〉〈 0,12 |+(1 − ,Bell )(| 12 + 〉〈 12 where 1 − ,Bell , 1 − 0 ,Bell and 1 − ,Bell are the detection errors of the singlet , non-polarized triplet 0 and the other Bell states ± in the partial Bell measurement. where ( 0 , ) is the probability of detecting ( 0 , ) in the Bell measurement without any detection error. Note that  Fig.4 ).
Finally, we predict the fidelity of the QT protocol using the obtained errors. We anticipate that our protocol could teleport an arbitrary input state coherently because the 2ns ramp time from (1,0,2) to (2,0,1) is sufficiently shorter than the measured dephasing time of 21 ± 1 ns in the device. In this paper, however, we can evaluate only the classical fidelity of the QT protocol because the prepared states of the input qubit are incoherent. We define the classical fidelity to be the We therefore conclude that the main error source of the classical infidelity is leakage to/from + in the preparation and detection process of the singlet state.
The imperfections of preparing and detecting the singlet arise partly from large Δ and large Δ between adjacent QDs. The large Δ induces spin mixing between the ground singlet and + , leading to the leakage of + . Δ between QD1 and QD2 is estimated to be 500 MHz from the ST oscillation in QD1 and QD2 similar to Fig. 2d and it induces the fast relaxation of | 0 〉 in the Bell measurement decreasing 0 ,Bell . While 0 ,Bell does not affect the classical fidelities ↑ and ↓ , it is important to the quantum mechanical fidelity in coherent teleportation of qubits. Although precise tunings of tunnel couplings can mitigate these problems, they may be avoided by redesigning the micromagnet in future experiments. For example, one can suppress Δ and Δ by using a micromagnet in a relatively symmetric geometry with respect to the QD array 32 while maintaining the strong slanting field for EDSR.
In summary, we demonstrate a simple and efficient protocol of the probabilistic QT of a spin qubit in a GaAs TQD device. The ground state initialization of a doubly occupied dot together with a simple pulsed control of detuning allows for the preparation of an entangled state as well as the Bell measurement. The statistics of spin polarization of the output qubit depends on the outcome of the Bell measurement and reproduces that of the input qubit, demonstrating that the spin orientation is teleported from the input qubit to the output qubit. Furthermore, considering our short operation time, we expect that our protocol could teleport an arbitrary input state. We find that the main error source in this protocol is the mixing of the entangled states with + substantially due to the large difference of local magnetic fields, which may be improved by optimizing the device design. Our demonstration is among the first demonstrations of teleportation with a single electron spin qubit in semiconductor quantum dots. Our results open a path to demonstrate quantum algorithms with three or more qubits in semiconductor electron spin qubits.

Supplementary material for Probabilistic teleportation of a quantum dot spin qubit
Y. Kojima et al.

I. Inter-dot tunnel couplings between adjacent QDs
It is assumed in QT that the output qubit is not directly coupled to the input qubit. In the TQD, there may be a direct exchange interaction between spins in QD1 and QD3 when large tunnel couplings between adjacent QDs are simultaneously turned on. To estimate the nearest-neighbor tunnel coupling, we measure the energy difference between the two eigenstates in the ST0 subspace, using a resonant SWAP technique. First we load a singlet in (2,0,1) ((1,0,2)) from the reservoir and ramp the detuning around the zero-detuning point (1,1,1) (in Fig. 1c) using slow adiabatic passage to initialize into |↑↓〉 ( |↓↑〉) 12 . Then, we instantaneously pulse to the operation point and modulate by applying MW to the PC gate to swap the two spins resonantly. The resonance frequency is equivalent to . Finally, we adopt the reversed gate pulse to distinguish |↑↓〉 and |↓↑〉 in the similar technique to PSB and measure the spin flip probability. The results of the detuning dependence of the probabilities are shown in Fig. S1.
To extract the dependence of , we fit the data in Fig. S1. We get the value of by taking the MW frequency where spin non-flip probability is minimum at a fixed detuning. We use = √ 2 + Δ 2 , where = However, the faintness and the difficulty in performing resonant SWAP empirically implies that the inter-dot tunnel coupling is much weaker than in QD1-QD2 and QD2QD3 can be ignored in the (1,1,1) region. Considering the operation time required for our QT is a few nanoseconds, which is much faster than the inverse of the adjacent exchange couplings, we conclude that direct interaction between QD1 and QD3 has no effect on our results. The yellow dashed line is the fitting result of . The MW burst time is 700 ns which is long enough to see the spectrum.

II. Estimation of the readout fidelity of input qubit
We can calculate the readout fidelities of input qubit from the signal separation between two charge states and the noise broadening of the charge sensor. Figure S2 is the histogram of single-shot readout result of PSB. It shows a bimodal distribution having a singlet peak at and a triplet peak at . The threshold voltage is chosen to maximize the averaged charge state readout fidelities. When the input spin state is spin-up (spin-down), the single-shot outcome of PSB is singlet (triplet). When we consider the model taking into account the relaxation of triplet states, we obtain ↑,in = 0.96 and ↓,in = 0.90. Fig. S2. The histogram of the PSB measurement results in the readout of Q1. The left population peak is for a singlet outcome (Q1 is spin-up) and the right is for triplet (Q1 is spin-down). The solid line is the fitting result using the model taking into account the relaxation of triplet states. We get = 0.02688, = 0.02784, σ = 0.0002508.

III. Estimation of the readout fidelity of output qubit
We use the spin-selective tunneling to the reservoir for readout of output qubit. In this readout method we judge the spin state by whether or not to detect the blip signal caused by electron tunneling events in the measurement time. To estimate the readout fidelity of the output qubit we use the model of Keith et al 30 . The error of the readout is decomposed into two factors. One is the error in the state-to-charge conversion (STC) and the other is the error in electrical detection of the blip signal. The readout fidelity f out,↑ and f out,↓ can be described by Here we denote by STC ↓ ( STC ↑ ) the probabilities that ↓ (↑) does (not) tunnel to the reservoir and denote by ↓ ( ↑ ) the probabilities that we do (not) detect a blip signal for where there is a (no) tunneling event.
To estimate STC ↑ and STC ↓ , we examine the tunnel rate OUT ↑(↓) from QD3 to the reservoir. We manipulate the spin in QD3 by EDSR with a frequency-chirped MW and realize the spin-selective tunneling between QD3 and the reservoir.
The distribution of the electron tunneling time gives the tunnel rate of each state. In our measurement condition, we find