## Main

Quantum teleportation is the central routine for reliably sending qubits across lossy network links3, as well as a key primitive of quantum network protocols and applications2,11,12. Using a teleporter in the form of a pre-shared entangled state, the quantum information is transferred by performing a joint Bell-state measurement (BSM) on the sender’s part of the entangled state and the qubit state to be teleported. The state is recovered on the receiving node by a gate operation conditioned on the BSM outcome3. Because the quantum information is not transmitted by a physical carrier, the protocol is insensitive to loss in the connecting photonic channels and on intermediate nodes. A deterministic BSM combined with real-time feed-forward enables unconditional teleportation, in which state transfer is achieved each time a qubit state is inserted into the teleporter.

Pioneering explorations of quantum teleportation protocols were performed using photonic states4,5,6. Following the development of quantum network nodes with stationary qubits, remote qubit teleportation was realized between trapped ions7, trapped atoms8,10, diamond nitrogen-vacancy (NV) centres9 and memory nodes based on atomic ensembles14.

Although future quantum network applications will widely use teleportation between non-connected nodes in the network, the demanding set of requirements on the pre-shared entanglement, the BSM and the coherence times for enabling real-time feed-forward has, so far, prevented the realization of teleportation beyond directly connected stationary network nodes.

Here we overcome these challenges by a set of key innovations and achieve qubit teleportation between non-neighbouring network nodes (see Fig. 1a). Our quantum network consists of three nodes in a line configuration, Alice, Bob and Charlie. Each node contains a NV centre in diamond. Using the NV electronic spin as the communication qubit, we are able to generate remote entanglement between each pair of neighbouring nodes. In addition, Bob and Charlie each use a nearby 13C nuclear spin as a memory qubit. The steps of the teleportation protocol are shown in Fig. 1b. To prepare the teleporter, we use an entanglement swapping protocol mediated by Bob, similar to a quantum repeater protocol15, to establish entanglement between Alice and Charlie. Once successful preparation of the teleporter is heralded, the input qubit state is prepared on Charlie and finally teleported to Alice.

## Entanglement fidelity of the network links

A key parameter for quantum teleportation is the fidelity of the pre-shared entangled state between Alice and Charlie. As we generate this state by entanglement swapping, its fidelity can be increased by mitigating errors on the individual links. Our network generates entanglement between neighbouring nodes using a single-photon protocol16,17 in an optical-phase-stabilized architecture18. The building block of this protocol is a qubit–photon entangled state created at each node. To generate this entangled state, we initialize the communication qubit in a superposition state $$|\psi \rangle =\sqrt{\alpha }|0\rangle +\sqrt{1-\alpha }|1\rangle$$ and apply a state-selective optical pulse that transfers the population from $$|0\rangle$$ to an optically excited state. Following spontaneous emission, the qubit state is entangled with the photon number (0 or 1 photon). We perform this protocol on both nodes and interfere the resonant photonic states on a beam splitter (Fig. 2a). Detection of a single photon in one of the output ports ideally heralds the generation of an entangled state $$|\psi \rangle =(|01\rangle \pm |10\rangle )/\sqrt{2},$$ in which the ± phase is set by the detector that clicked. Figure 2b shows the joint outcomes of qubit measurements in the computational basis after entanglement is heralded, showing the expected correlations.

The infidelity of the generated state has three main contributions: double $$|0\rangle$$ state occupancy, double optical excitation and finite distinguishability of the photons18,19. In the case of double $$|0\rangle$$ state occupancy (which occurs with probability α), both communication qubits are in the $$|0\rangle$$ state and have emitted a photon. Detection of one of these photons leads to false heralding of an entangled state. The second effect, double excitation, is due to the finite length of the optical pulse compared with the optical lifetime of the emitter. There is a finite chance that the communication qubit emits a photon during this pulse, is subsequently re-excited and then emits another photon, resulting in the qubit state being entangled with two photons. Detection or loss of the first photon destroys the coherence of the qubit–photon entangled state and detection of the second photon can then falsely herald the generation of an entangled state.

Crucially, false heralding events caused by double $$|0\rangle$$ state occupancy and double excitation are both accompanied by an extra emitted photon. Therefore, detection of this extra photon allows for unambiguous identification of such events and thus for real-time rejection of false heralding signals. We implement this rejection scheme by monitoring the off-resonant phonon-side band (PSB) detection path on both setups during and after the optical excitation (see Fig. 2a).

To investigate the effect of this scheme, we generate entanglement on the individual links and extract the entanglement heralding events for which the PSB monitoring flagged the presence of an extra photon. For these events, we analyse the corresponding qubit measurements in the computational basis (Fig. 2c).

We identify two separate regimes: one during the optical pulse (purple) and one after the optical pulse (yellow). When a photon is detected on Alice’s (Bob’s) PSB detector during the optical pulse, we see that the outcome 01 (10) is most probable (purple data in Fig. 2c), showing that only one setup was in the $$|0\rangle$$ state and thus that both detected photons originated from Alice (Bob). The detection of PSB photons during the optical pulse thus primarily flags double excitation errors. By contrast, when a photon is detected after the optical pulse in either Alice’s or Bob’s PSB detector, the outcome 00 is most probable (yellow data in Fig. 2c), indicating that both setups were in the $$|0\rangle$$ state and emitted a photon. PSB photon detection after the optical pulse thus flags the double $$|0\rangle$$ state occupancy error. We find similar results to Fig. 2c for the entangled states generated on the Bob–Charlie link; see Extended Data Fig. 2. The improvement in fidelity from rejecting these false heralding events in our experiment is set by the combined probability of occurrence (≈9%; see Supplementary Information) multiplied by the probability to flag them (given here by the total PSB photon detection efficiency of ≈10%).

The third main source of infidelity, the finite distinguishability, can arise from frequency detunings between the emitted photons20. Whereas most of these detunings are eliminated up front by the charge-resonance (CR) check before the start of the protocol (see Supplementary Information), the communication qubits may still be subject to a small amount of spectral diffusion. In our single-photon protocol, this leads to dephasing that is stronger for photons that are detected later relative to the optical pulse. By shortening our detection window, we can increase the fidelity of the entangled state at the expense of a lower entangling rate. For the experiments below (unless mentioned otherwise), we use a detection window length of 15 ns. Figure 2d summarizes the measured improvements on the individual links and the estimated effect on the Alice–Charlie entangled state fidelity. The increase of ≈3% is instrumental in pushing the teleportation fidelity above the classical bound.

## Memory qubit coherence

In the preparation of the teleporter, we reliably preserve the Alice–Bob entangled link on the memory qubit, by aborting the sequence and starting over when the Bob–Charlie entangled state is not heralded within a fixed number of attempts, the timeout.

The 13C memory qubits can be controlled with high fidelity by means of the communication qubit, although they can be efficiently decoupled when no interaction is desired. Recent work showed that, in a magnetic field of 189 mT, entanglement generation attempts with the communication qubit do not limit the memory dephasing time $${T}_{2}^{\ast }$$ (ref. 18), opening the door to substantially extending the memory preservation time with active coherence protection from the spin bath21. We realize this protection by integrating a decoupling π-pulse on the memory qubit into the experimental sequence that follows a heralding event, while ensuring that all phases that are picked up owing to the probabilistic nature of the remote entangling process are compensated in real time (Fig. 3a).

In Fig. 3b, we check the performance of this sequence by storing a superposition state on the memory qubit and measuring the Bloch vector length. We observe that, without the decoupling pulse, the decay of the Bloch vector length is not altered by the entanglement attempts, in line with previous findings18. By contrast, when we apply the decoupling pulse, the decay is slowed down by more than a factor of 6, yielding a N1/e decay constant of ≈5,300 entanglement attempts, the highest number reported so far for diamond devices. The difference in the shape of the decay indicates that intrinsic decoherence is no longer the only limiting factor. The improved memory coherence enables us to use a timeout of 1,000 entangling attempts, more than double that of ref. 18, which doubles the entanglement swapping rate.

High-fidelity memory qubit readout is required both in the preparation of the teleporter (at Bob) and during the teleportation protocol itself (at Charlie). The memory qubit is read out by mapping its state onto the communication qubit using quantum logic followed by single-shot readout of the communication qubit using state-dependent optical excitation and detection22. Owing to limited photon collection efficiency (≈10%) and finite cyclicity of the optical transition (≈99%), the communication qubit readout fidelity is different for $$|0\rangle$$ and $$|1\rangle$$ and the probability that the correct state was assigned is much larger if one or more photons were detected (assigned outcome 0) than if no photons were detected (assigned outcome 1)23. In previous work, we circumvented this issue by conditioning on obtaining the outcome 0 (ref. 18). However, this approach scales unfavourably, as it forces the protocol to prematurely abort with probability >50% at each memory qubit readout.

We resolve this challenge by introducing a basis-alternating repetitive readout for the memory qubit (see Fig. 3c). The key point of this readout strategy is, in contrast to earlier work24, to alternatingly map the computational basis states of the memory qubit to the communication qubit state $$|0\rangle$$ Figure 3d shows the readout fidelities of the nth readout repetition for the two initial states for the memory qubit on Bob (for Charlie, see Extended Data Fig. 3). We clearly observe the expected alternating pattern owing to the asymmetry of the communication qubit readout fidelities. Notably, the readout fidelity decays only by ≈1% per readout, showing that the readout is mostly non-demolition and several readouts are possible without losing the state.

## Teleporting qubit states from Charlie to Alice

With all innovations described above implemented, we perform the protocol as shown in Fig. 4a. First, we generate entanglement between Alice and Bob and store Bob’s part of the entangled state on the memory qubit using a compiled SWAP operation. Second, we generate entanglement between Bob and Charlie, while preserving the first entangled state on the memory qubit with the pulse sequence as described in Fig. 3a. Next, we perform a BSM on Bob followed by a CR check. We continue the sequence if the communication qubit readout yields outcome 0, the memory qubit readout gives a consistent outcome pattern and the CR check is passed. At Charlie, we perform a quantum gate that depends on the outcome of the BSM and on which detectors clicked during the two-node entanglement generation. Next, we swap the entangled state to the memory qubit. At this point, the teleporter is ready and Alice and Charlie share an entangled state with an estimated fidelity of 0.61.

Subsequently, we generate the qubit state to be teleported, $$|\psi \rangle ,$$ on Charlie’s communication qubit and run the teleportation protocol. First, a BSM is performed on the communication and memory qubits at Charlie. With the exception of unconditional teleportation (discussed below), we only continue the sequence when we obtain a 0 outcome on the communication qubit, when we have a consistent readout pattern on the memory qubit and when Charlie passes the CR check. The outcomes of the BSM are sent to Alice and, by applying the corresponding gate operation, we obtain $$|\psi \rangle$$ on Alice’s side.

We teleport the six cardinal states $$(\pm {\rm{X}},\pm {\rm{Y}},\pm {\rm{Z}}),$$ which form an unbiased set25, and measure the fidelity of the teleported states to the ideally prepared state (Fig. 4b). We find an average teleported state fidelity of F = 0.702(11) at an experimental rate of 1/(117 s). This value exceeds the classical bound of 2/3 by more than three standard deviations, thereby proving the quantum nature of the protocol. We note that this value provides a lower bound to the true teleportation fidelity, as the measured fidelity is decreased by errors in the preparation of the qubit states at Charlie (estimated to be 0.5%; see Supplementary Information).

The differences in fidelity between the teleported states arise from an interplay of errors in different parts of the protocol that either affect all three axes (depolarizing errors) or affect only two axes (dephasing errors). These differences are qualitatively reproduced by our model (grey bars in Fig. 4b). In Fig. 4c, we plot the teleportation fidelity for each possible outcome of the BSM. Owing to the basis-alternating repetitive readout, the dependence on the second bit (from the memory qubit readout) is small, whereas for the first bit (communication qubit readout), the best teleported state fidelity is achieved for outcome 0, due to the asymmetric readout fidelities. We also analyse the case in which no feed-forward is applied at Alice (see Methods); as expected, the average state fidelity reduces to a value consistent with a fully mixed state (fidelity F = 0.501(7)), emphasizing the critical role of the feed-forward in the teleportation protocol.

Finally, we demonstrate that the network can achieve unconditional teleportation between Alice and Charlie by using the BSM in a deterministic fashion. To this end, we revise the protocol at Charlie to accept both communication qubit outcomes, use all memory qubit readout patterns, including the inconsistent ones, and disregard the outcome of the CR check after the BSM. Using this fully deterministic BSM lowers the average teleportation fidelity by a few percent (Fig. 4d). At the same time, shortening the detection windows of the two-node entanglement generation is expected to yield an improvement in the fidelity, as discussed above. We find that, indeed, the average unconditional teleportation fidelity increases with shorter window lengths, reaching F = 0.688(10) for a length of 7.5 ns and a rate of 1/(100 s); see Extended Data Fig. 4. The current quantum network is thus able to perform teleportation beyond the classical bound, even under the strict condition that every state inserted into the teleporter be transferred.

## Outlook

In this work, we have realized unconditional qubit teleportation between non-neighbouring nodes in a quantum network. The innovations introduced here on memory qubit readout and protection during entanglement generation, as well as the real-time rejection of false heralding signals, will be instrumental in exploring more complex protocols2,11,12,13,26. Also, these methods can be readily transferred to other platforms, such as the group IV colour centres in diamond, the vacancy-related qubits in SiC and single rare-earth ions in solids27,28,29,30,31,32,33.

The development of an improved optical interface for the communication qubit34 will increase both the teleportation protocol rate and fidelity. Because of the improved memory qubit performance reported here, the network already operates close to the threshold at which nodes can reliably deliver a remote entangled state while preserving previously stored quantum states in their memory qubits. With further improvements, for instance, by integrating multi-pulse memory decoupling sequences21 into the entanglement generation, demonstration of deterministic qubit teleportation (with no pre-shared entangled state) may come within reach, which opens the door to exploring applications that call the teleportation routine several times. In addition, future work will focus on further improving the phase stabilization and extending the current schemes for use in deployed fibre35.

Finally, by implementing a recently proposed link layer protocol36, qubit teleportation and applications making use of the teleportation primitive may be executed and tested on the network through platform-independent control software, an important prerequisite for a large-scale future network.

## Methods

### Experimental setup

The basics of the experimental setup are described in ref. 18. In the current experiment, Charlie has access to a carbon-13 nuclear spin that acts as a memory qubit. The parameters used for the memory qubits of Bob and Charlie can be found in Extended Data Table 3. Furthermore, we have set up a classical communication channel between Charlie and Alice, such that Charlie can directly send the results of the BSM to Alice.

### Temporal selection of heralding photons

To eliminate any reflected excitation light in the heralding detectors, we make use of a cross-polarization scheme and perform temporal selection of the detected photons as described in ref. 37. We start the detection windows 4 ns (5 ns) after the highest intensity point of the excitation pulse, for the AB (BC) entangled link, to ensure sufficient suppression of excitation laser light in the detection window.

### Memory qubit coherence Bob

We use the sequence described in Fig. 3a to preserve the state of the memory qubit during entanglement attempts. To characterize the decoupling sequence, we compare it to the sequence in which we do not apply the decoupling pulse on the memory qubit and/or the sequence in which we idle instead of performing entanglement attempts. We characterize the coherence of the memory qubit by storing the six cardinal states. We average the results for the eigenstates $$(|0\rangle ,|1\rangle )$$ and superposition states $$(|\pm {\rm{X}}\rangle {\rm{and}}|\pm {\rm{Y}}\rangle ).$$ In Extended Data Fig. 5a, we plot the Bloch vector length $$b=\sqrt{{b}_{x}^{2}+{b}_{y}^{2}+{b}_{z}^{2}},$$ with bi the Bloch vector component in direction i.

Over the measured range, the eigenstates show little decay. The decay of the superposition states is fitted with the function $$f(x)=A{{\rm{e}}}^{-{(x/{N}_{1/{\rm{e}}})}^{n}}.$$ The fitted parameters can be found in Extended Data Fig. 5b.

The use of the decoupling pulse πM on the memory qubit increases the N1/e by more than a factor of 6. Moreover, the initial Bloch vector length A is higher with the πM pulse. This is mainly explained by the second round of phase stabilization18 in between swapping the state onto the memory qubit and starting the entanglement generation process. The phase stabilization takes ≈350 μs and, during this time, the memory qubit is subject to intrinsic $${T}_{2}^{\ast }$$ dephasing, which can be efficiently decoupled using the πM pulse.

### Communication qubit coherence

In various parts of the protocol, we decouple the communication qubits from the spin bath environment to extend their coherence time. On Alice, we start the decoupling when the first entangled link is established and stop when the results of the BSM to teleport the state are sent by Charlie. On Bob, we decouple the communication qubit when the memory qubit is being rephased. On Charlie, the communication qubit is decoupled from the point that entanglement with Bob is heralded up to the point at which Bob has finished the BSM, performed the CR check and has communicated the results. All these decoupling times are dependent on how many entanglement attempts are needed to generate the entangled link between Bob and Charlie.

We characterize the average state fidelities for different decoupling times; see Extended Data Fig. 6a. We investigate eigenstates and superposition states separately. We fit the fidelity with the function $$f(t)=A{{\rm{e}}}^{-{(t/{\tau }_{{\rm{coh}}})}^{n}}+0.5.$$ The fitted parameters are summarized in Extended Data Fig. 6b. For each setup, the minimum and maximum decoupling times used are indicated by the shaded regions in Extended Data Fig. 6a. The left-most border is the decoupling time when the first entanglement attempt on Bob and Charlie would be successful and the right-most border is when the last attempt before the timeout of 1,000 attempts would herald the entangled state.

### Model of the teleported state

A detailed model of the teleported state can be found at https://doi.org/10.4121/16645969. The model comprises elements from ref. 18 and is further extended for the teleportation protocol. We take the following noise sources into account:

• Imperfect Bell states between Alice and Bob, and between Bob and Charlie.

• Dephasing of the memory qubit of Bob during entanglement generation between Bob and Charlie.

• Depolarizing noise on the memory qubits of Bob and Charlie, owing to imperfect initialization and swap gates.

• Readout errors on the communication qubits of Bob and Charlie and readout errors on the memory qubits of Bob and Charlie when using the basis-alternating readout scheme, which result in incorrect feed-forward gate operations after the BSMs.

• Depolarizing noise on Alice during the decoupling sequence.

• Ionization probability on Alice.

An overview of the input parameters and the effect of the different error sources are given in Extended Data Table 4.

### Calculation of teleported state fidelity without feed-forward operation

In Fig. 4c, we show the fidelity of the teleported state in case no feed-forward operations would have been applied on Alice. To extract this data, we follow the same method as in ref. 9. We perform classical bit flips on the measurement outcomes to counteract the effect of the feed-forward gate operations (as if the gate was not applied) for each BSM outcome. We do this for all six cardinal states and compute the average fidelity. We assume the errors of the gate in the feed-forward operations to be small.