Chip-to-Chip High-Dimensional Teleportation via A Quantum Autoencoder

Quantum teleportation transfers unknown quantum states from one node in a quantum network to another. It is one of the crucial architectures in quantum information processing. The teleportation of high-dimensional quantum states remains challenging due to the difficulties in executing high-dimensional Bell state measurement. Here, we propose a Quantum Autoencoder-Facilitated Teleportation (QAFT) protocol for high-dimensional quantum teleportation, and report the first demonstration of QAFT on qutrits using an integrated photonic platform for future scalability. The key strategy is to reduce the dimension of the input states by erasing redundant information and reconstruct its initial state after chip-to-chip teleportation. Machine learning is applied in training the autoencoder to facilitate the teleportation of any state from a particular high-dimensional subspace and achieve the reconstruction of the unknown state (by the decoder) with high fidelities (~0.971). Experimentally, we teleport unknown qutrits by generating, transferring and manipulating photons, and training quantum autoencoders on a silicon chip. A teleportation fidelity of ~0.894 is demonstrated. Our scheme opens pathway towards quantum internet and cryptography to transfer unmeasured states in a quantum computer. It also lays the groundwork for machine learning technologies in quantum networks and quantum computations.


Introduction
Data compression is a ubiquitous processing that we use daily to transfer large audio and video files. Autoencoder serves as an efficient means for large dataset compression 1, 2 , by learning an encoder map from an input data to a latent space and a decoder map from the latent space to the input space, over a training set. Moving to the quantum regime, there are some fundamental differences between classical and quantum information. The amount of information that one can extract from a classical system matches the amount of information needed for a complete description of the state of the system. However, in a quantum system, the complete description of a quantum state strictly requires an infinite amount of information. Nevertheless, one can devise quantum compression protocol based on Schur-Weyl transformation 3 . The quantum autoencoder [4][5][6][7] is also proposed to efficiently compress a particular dataset of quantum states using machine learning, and its concept has been demonstrated using bulk optics 8, 9 . Through data compression, quantum autoencoder can become a useful tool for transferring high-dimensional quantum information between remote parties (e.g., chip-tochip teleportation), reducing the quantum memory, quantum communication channels and the size of quantum gates. Quantum teleportation [10][11][12] is a crucial protocol for the physical implementation of many quantum communication and quantum computation schemes, including quantum relays 13 , quantum repeaters 14 and linear optics quantum computing 15,16 .
Quantum teleportation enables the moving of a qubit from one place to another without the real need to physically transport the underlying particle where the qubit is attached. To date, quantum teleportation has been experimentally generalized by different physical systems such as photons [17][18][19][20][21][22][23] , optical coherent states 24 , atoms [25][26][27][28] and nuclear magnetic states 29 .
However, most controllable teleportation schemes only consider two-dimensional quantum systems, and the teleportation of high-dimensional quantum states still remains challenging.
Bennett et al. 10 first proposed to realize the teleportation of an unknown d-dimensional state by replacing the maximally entangled state of two qubits with that of two entangled qudits, and by replacing the Bell measurement with a generalized measurement involving a set of maximally entangled orthonormal or mutually unbiased bases. Besides the difficulty in generating high-quality qudit entanglement [30][31][32][33][34] , the proposal faces a major obstacle in implementing high-dimensional Bell State measurement (high-D BSM). Due to the linear optical limitation 35 , the d 2 Bell states in a complete orthonormal basis cannot be identified, except for = 2. Several theoretical approaches have been developed [36][37][38][39] .
Recently, two experiments (not integrated) were reported 40,41 to teleport a qutrit-by-path.
By adopting ancillary photons, Luo et al. 40 observe specific click patterns of three interfering photons that indicate successful projections into one of the 9 Bell states, at a success probability of 1/81; Hu et al. 41 observe the coincidence events of six photons and use non-maximally entangled state, achieving a success probability of 1/18. Both works exhibit high quality of their proposals, though some concerns present in their scalability, i.e., for a d-dimensional system, they require ( − 2) ancillary photons and ⌈ + ( ) − 1⌉ pairs of ancillary photons, respectively. The increment of dimensionality would result in the demand for more ancillary photons, and a decline of success probability.
Here, we propose a quantum-autoencoder-facilitated teleportation (QAFT) protocol, which automatically compresses the high-dimensional quantum states into qubits, further teleports and decodes them at the receiver's end. This protocol does not require ancillary photons and has a success probability of 1/2. We report, for the first time, a silicon photonic chip implementation of quantum autoencoder, and the complete integration and demonstration of QAFT on qutrits. The generation, teleportation and measurement of photon states, as well as the training of encoder are all performed on the silicon chip. Unlike previous universal schemes, our protocol requires separate encoders for separate subspaces.
Nevertheless, we are capable of training a universal encoder for any particular subspace of qutrits by taking a finite sample of these qutrits randomly and learning to compress them into qubits. During training, single-shot measurement is carried out to ensure that no qutrit is copied or measured twice. The quantum autoencoder achieves almost lossless compression on the qutrit to qubit, with a reconstruction fidelity of ~0.971. After training, we can teleport any further states and reconstruct them at the receiver chip. A high teleportation fidelity of ~0.894 between the input qutrit and the teleported qutrit is achieved.
The integrated silicon photonics technology endows the implementation of QAFT with high stability and easier scalability. Our scheme will come in handy for quantum internet, cryptography and transferring quantum states by reducing the requirements on quantum memory, quantum communication channels and the size of quantum gates.

The QAFT Protocol and Chip Design
The overarching idea of QAFT is that by training an autoencoder, the input states can be compressed into qubits, teleported, and reconstructed by the decoder as shown in Figure   1a. The transmitter and the receiver each hold half of an Einstein-Podolsky-Rosen (EPR) pair. At the transmitter, the initial qutrit is compressed into qubit by the trained encoder, and the BSM is performed. Depending on the result of BSM and the settings of encoder, the receiver will set up the decoder and reconstruct the initial qutrit from the teleported qubit. The training of the autoencoder is crucial to the success of teleportation. are encoded as dual-rail qubits. The two signal photons become one qubit, and two idler photons become the other by passing the crossing structure. The two qubits are maximally entangled in the form of Bell state | 3 ⟩ with fidelities of 0.960 ± 0.004 (see Fig. S3e), of which one qubit is held by the transmitter and the other is held by the receiver. The idler photon of the bottom pair is used to generate a path-encoded qutrit using the linear optical quantum circuit and its signal photon is used for heralding in measurement. The qutrit is compressed by a well-trained encoder. At the transmitter, a Bell state measurement of the EPR qubit and the teleportee qubit is performed, yielding one of the four measurement outcomes, which is actually one of the two possible measurement outcomes | 3 ⟩ and | 6 ⟩ in our design, while | 3 ⟩ and | 6 ⟩ are indistinguishable by the measurement results. The key component is the reprogrammable two-qubit operator, which entangles two qubits (previously never interacted) in Bell projection. The measurement results can be encoded by two classical bits of information.
The EPR qubit is transmitted along a long distance to the receiver chip through the polarization rotator and splitter (PRS) that converts the path state in waveguide to the horizontal (vertical) state in optical fiber (see Supplementary Note 6). The two-bit BSM results, and the encoder settings are transmitted from the transmitter chip to the receiver chip through classical information channel. The EPR qubit held by the receiver will be identical to the teleportee qubit if the BSM result is | 3 ⟩. Alternatively, it can be fixed up if the BSM result is | 6 ⟩. Once the qubit is teleported, the decoder integrated on the receiver chip will reconstruct it to the initial qutrit. As the encoder is a unitary transformation, the decoder, which is integrated in the receiver chip, is set up as the inverse of the encoder. The false-colour micrograph of the fabricated chip is shown in Figure 1c  An evolutionary optimization algorithm is utilized to iteratively update parameters of the encoder, aiming to compress a subspace of qutrits. Given a training qutrit, the encoder can compress the qutrit into a qubit (that is pending for teleportation), while leaving the trash mode unoccupied. Any probability of finding photons in the trash mode implies an imperfect compression and will result in the failure to reconstruct the initial qutrit when the trash mode is discarded. The photon occupancy is estimated over a series of arbitrary input qutrits via single-shot measurement, i.e., any qutrit will not be copied or sent twice, and the training algorithm will never measure the same qutrit twice because, practically, the qutrit generator does not know what state it produces and the sequence of output states.
After training, any further states can be teleported and reconstructed at the receiver's end.

Results and Discussions
Training of quantum autoencoder with single-shot measurements. Figure 3a shows the flow chart of the training process. The main components are the individually controllable photon activator, qutrit generator ( 96+ and 96: ) and trainable encoder ( 96: , <6= and <6> ). Random qutrits are generated by a rotation matrix that maps random qubits to qutrits such that they belong to the same subspace and can be compressed by a common encoder. An evolutionary genetic algorithm is adopted to update the free parameters of the unitary encoder iteratively. The training starts from the population initialization. Each population has 20 individuals, and each individual AAAAA⃗ = ( 9 , + , … F ) is composed of 8 free parameters, which are the electrical currents applied on the 8 phase shifters, forming arbitrary unitary matrices. As an example, when the AAAAA⃗ in the initial population is applied to the encoder, a random qutrit will be generated at once for the evaluation of this individual. A total of 50 random qutrits are used to estimate the photon occupancy with each qutrit measured only once. Each time after generating and measuring a qutrit, we will randomize the qutrit generator such that the next qutrit is a different random qutrit. The trash mode of the encoder is monitored by the counting logic, which outputs the required time (i.e., ∆ ) until the first click observed at the trash mode, which is inversely proportional to its photon occupancy (longer ∆ leads to lower photon occupancy). If ∆ is as long as the time required to detect the dark noise (which is denoted as = 10 9L ps), we can regard the photon occupancy of the trash mode as close to zero. Accordingly, we design the fitness function of the genetic algorithm as ∝ 1 − 96∆P/R , and the training objective as → 0. So far, the fitness evaluation of the individual AAAAA⃗ is achieved. Subsequently, the same process is repeated for the remaining individuals in current population. If the fitness values satisfy the stopping criteria, the best individual in current population is returned as the optimal solution. Otherwise, the algorithm continues and the individuals in current population are transformed by the genetic operators (i.e., selection, crossover, and mutation) to generate the offspring (a population of 20 new individuals). The offspring will be re-evaluated until the algorithm converges. Figure 3b shows the timing diagram of the counting logic. Two signal channels, a laser channel and a photon channel (monitored at the trash mode) are connected to the counting logic. We create a gate that starts at one of the laser pulses with a period of 10 10 ps, which is the minimum required time to detect the dark noise. If no photon signal is detected during this period, the photon signal is regarded as a dark noise. The gated stream in recording is composed of clicks in both laser channel and photon channel. The output of the time tagger is the time difference between the first photon click and the first laser click, i.e., ∆ = TUVPVW − XY/Z[ . The photon clicks after the first click will be discarded. If a photon click appears before the first laser click, the photon will also be discarded.  Fig. S2). As seen from the tomography results of the initial qutrits and the compressed states (Fig. S2), the device achieves a high-quality compression from initial qutrits to the compressed states, reducing the occupation probability of the trash mode to 0.023 ± 0.011. When we reconstruct the qutrit by using the decoder (i.e., the inverse of the trained encoder), the fidelity = ⟨ | | ⟩ between the initial qutrit | ⟩ and the reconstructed density matrix is reported to be 0.971 ± 0.013.

Bell Projection and teleportation of qutrits.
In the teleportation protocol, an unknown quantum state can be transmitted to another location by locally collapsing the state and remotely reconstructing it. This requires the access to Bell states and Bell measurements.
A Bell projector is employed to entangle initially separable qubits and measure qubits in the Bell basis. The schematic diagram for the Bell projector devised for dual-rail qubits is depicted in Figure 1b (and Fig. S8). We denote the three generated photons from top to bottom as "the 1 st qubit", "the 2 nd qubit", and "the 3 rd qubit", and denote the four detectors of the BSM from top to bottom as "D1", "D2", "D3" and "D4". By deliberately designing the circuitry, k lZXX can distinguish the Bell states | ⟩ ± from the others, as shown in Table   S1. We distinguish | ⟩ 3  Finally, we demonstrate the QAFT with the chip-to-chip teleportation by teleporting several randomly generated qutrits. The initial arbitrarily prepared qutrit is compressed by a well-trained encoder to a qubit, and the qubit is teleported from the 3 rd qubit to the 1 st qubit. Then, at the receiver chip, the teleported 1 st qubit is reconstructed to the qutrit by the decoder. The decoder is built based on the encoder, whereby the information of which is classically transmitted from the transmitter chip to the receiver chip. The mean fidelity between the density matrices of the initial qutrit and the transported qutrit is + ttt =0.894 ± 0.026 as shown in Figure 4b. One of the reasons of the degradation in the teleportation fidelity from 9 s to + ttt can be attributed to the reconstruction ability of the encoder.
Nevertheless, the high fidelity proves that the QAFT can successfully compress the input qutrits using a trained encoder and reconstruct them through the decoder after a longdistance teleportation.

Conclusion
We propose and demonstrate a QAFT protocol on a silicon photonic quantum chip that teleports arbitrarily generated qutrits by training a quantum autoencoder that compresses the input qutrits to qubits. The proposal is generic, and it is possible to extend the scheme where is the phase shifter. According to the measured fringe of MZI for PRS on the encoder and decoder chips, the visibilities Ven = 3.0% and Vde = 5.6% are achieved respectively, and the ratio is estimated to be 0.97. To couple the two propagation modes simultaneously, we adopt the spot size converter (SSC) as the polarization-independent edge coupling structure. The one-dimensional grating coupler is not adoptable here because it rejects the TM mode at an extinction ratio of ~20 dB. A polarization controller is placed between the transmitter chip and the receiver chip, and the polarization alignment of the two chip is done by sending |0⟩ and |1⟩ as the calibration references.
Data availability. The data that support the findings of this study are available from the corresponding authors on reasonable request.