Resource-efficient high-dimensional subspace teleportation with a quantum autoencoder

Abstract

Quantum autoencoders serve as efficient means for quantum data compression. Here, we propose and demonstrate their use to reduce resource costs for quantum teleportation of subspaces in high-dimensional systems. We use a quantum autoencoder in a compress-teleport-decompress manner and report the first demonstration with qutrits using an integrated photonic platform for future scalability. The key strategy is to compress the dimensionality of input states by erasing redundant information and recover the initial states after chip-to-chip teleportation. Unsupervised machine learning is applied to train the on-chip autoencoder, enabling the compression and teleportation of any state from a high-dimensional subspace. Unknown states are decompressed at a high fidelity (~0.971), obtaining a total teleportation fidelity of ~0.894. Subspace encodings hold great potential as they support enhanced noise robustness and increased coherence. Laying the groundwork for machine learning techniques in quantum systems, our scheme opens previously unidentified paths toward high-dimensional quantum computing and networking.

Fig. 1.. The QAFT protocol in high dimensions.

(A) Overview of the QAFT protocol. The transmitter and the receiver each hold one half of an EPR pair. An initial qutrit is compressed to a qubit by the trained encoder. The transmitter interacts the qubit with the half of an EPR pair and measures the two qubits in its possession. The BSM results and encoder settings are transmitted to the receiver via the classic channel. The decoder reconstructs the initial qutrit accordingly. (B) Design of the QAFT chip. The transmitter chip consists of a nondegenerate photon generator, entanglement, encoder, and Bell projector. The receiver chip integrates the decoder. The two subchips are coherently linked by a single mode fiber. At the transmitter, the qutrit is compressed by the encoder to a qubit, and Bell measurements are performed. At the receiver, the decoder reconstructs the initial qutrit state from the teleported qubit. p, pump; s, signal; i, idler. (C) False-color micrograph of the QAFT chip. The transmitter and receiver are integrated to form a two-way transceiver chip. Output photons (D1 to D6) are coupled by spot-size converters (SSCs) and detected off-chip by superconducting single-photon detectors. During the training of the encoder, another single-photon detector (D7) is placed at the trash mode to observe the photon occupancy. B1 and B2 are connected by a 10-m fiber, and a polarization rotator and splitter (PRS) are used to convert the path state in waveguide to the horizontal (vertical) state in optical fiber.

Fig. 2.. Implementation of the trainable quantum autoencoder on a photonic chip.

(A) Graphical representation of a 3-2-3 quantum autoencoder. Interconnected nodes represent space dimensions, and edges connecting adjacent layers represent the weights (a unitary transformation). The encoder shrinks high-dimensional data into lower-dimensional space, and then the decoder expands the space to reconstruct the original input states should the compression is lossless. (B) Architecture of the quantum autoencoder that consists of an encoder (E) and a decoder (D). The two modes that pass to the decoder are “qubit modes,” and the one that is not used is “trash mode.” The encoder is trained to retain all information on qubit modes while minimizing the occupation probability of the trash mode. The training parameters on the quantum circuit are optimized by a machine learning algorithm, according to the photon occupancy of the trash mode. (C) On-chip training process and the trainable encoder circuit. The qutrit generator (with five free parameters θ₁₋₂ and ϕ₁₋₃) and the trainable encoder (with eight free parameters α₁₋₃, θ₄₋₆, and ϕ₄₋₅) are independently controlled. MZIs with different functionalities are marked by different colors, including the qutrit preparation (blue), encoder (red), and qutrit state tomography (gray). The decoder is the inverse of the encoder. The implementation of an evolutionary cycle involves the population initialization, the individual evaluation, and genetic operators (i.e., selection, crossover, and mutation). Individuals are evaluated by their photon occupancy at trash mode over randomly generated qutrits.

Fig. 3.. Single-shot training of the quantum autoencoder.

(A) Flowchart of single-shot training, which starts from population initialization. Individuals are evaluated on chip by the photon occupancy, which is inversely proportional to the time taken (i.e., ∆t) for the first click observed at trash mode. (B) Timing diagram of the counting logic. Two signal channels, the laser channel and the photon channel (at the trash mode), are monitored. A gate with a period of T = 10¹⁰ ps (i.e., the time required to detect dark noise) is created from one of the laser pulses. The time difference ∆t between the first photon click and the first laser click is returned as the output. The training objective is Δt → T, so that the fitness f ∝ 1 − e^∆t/T can approach f = 0. (C) The evolution of the time spent until the first click at the trash mode, which intuitively shows the convergence. Most individuals in the final generation achieve photon occupancy comparable to dark noise. (D) Evolution of the best and average fitness value. The fitness value of the final best encoder is 0.016, and the average fitness of the final population is 0.082. (E and F) Contrast of the time taken for the first click and fitness value in the initial and final generation. (G) Statistics of the 20 individuals in the initial and the final generation, respectively. Most individuals in the final generation have almost the same current value, with an average SD of 0.016 mA.

Fig. 4.. BSM and quantum state tomography results for chip-to-chip teleportation.

(A) Chip-to-chip teleportation results. A total of six elementary states were teleported from the transmitter to the receiver, respectively. The density matrices of the six states were constructed by full quantum state tomography on the receiver chip. Reconstructed density matrices are shown along with the measured fidelities, reporting a mean fidelity of 0.914 ± 0.022. (B) Qutrit encoding and decoding. The teleported qubit was decoded to reconstruct the initially generated qutrit on the receiver chip. The decoder is built according to the classical information transmitted to the receiver chip. Full quantum state tomography was applied to reconstruct the density matrix and report a mean fidelity of 0.894 ± 0.026. All error bars refer to ±1 SD estimated from Poissonian photon-counting statistics through Monte-Carlo simulation.