Experimental Realization of a Quantum Autoencoder: The Compression of Qutrits via Machine Learning

With quantum resources a precious commodity, their efficient use is highly desirable. Quantum autoencoders have been proposed as a way to reduce quantum memory requirements. Generally, an autoencoder is a device that uses machine learning to compress inputs, that is, to represent the input data in a lower-dimensional space. Here, we experimentally realize a quantum autoencoder, which learns how to compress quantum data using a classical optimization routine. We demonstrate that when the inherent structure of the dataset allows lossless compression, our autoencoder reduces qutrits to qubits with low error levels. We also show that the device is able to perform with minimal prior information about the quantum data or physical system and is robust to perturbations during its optimization routine.

Figure 1

(a) The concept of an autoencoder. Through an encoding process ($E$), autoencoders represent data in a lower-dimensional space; if the compression is lossless, the original inputs can be perfectly recovered through a decoding process ($D$). (b) The scheme of our qudit-based autoencoder, equivalent to the gray shaded section in (a), for the case of a compression of qutrits to qubits. A unitary transformation $U$ characterized by a set of parameters converts between three input modes and three output modes. The iterative training of the parameters aims to minimize the occupation probability of the third output mode, the “junk” mode, across a set of training input states. Lossless compression is achieved when the junk mode is unoccupied. Given that the encoding step is performed with a unitary transformation, the decoding step would simply be the inverse of the unitary, using a vacuum state in the third input mode.

Figure 2

Experimental layout. (a) Single photon source. We use a 410 nm continuous-wave diode laser to pump a beta barium borate (BBO) crystal to produce photon pairs via type-I spontaneous parametric down-conversion. One photon is collected and detected to herald the second photon; this second photon is fiber coupled and passes through a fiber polarization controller (FPC). (b) State preparation. First, a qubit is encoded in the polarization state of the photon via a motorized half-wave plate (HWP) and quarter-wave plate (QWP). Then the photon passes through a polarizing beam displacer. In the lower spatial mode, a random but fixed birefringent element (a randomly oriented wave plate for a different wavelength) scrambles the polarization, mapping vertical polarization to some superposition of horizontal and vertical. The top spatial mode has matching optical elements (not shown) set to the optic axis, in order to match the optical path length to within the photon’s coherence length. By changing the motorized wave plates, sets of qutrits are created in polarization and path modes in such a way that they are compressible into qubits. (c) Autoencoder. A $3\times 3$ unitary transformation with four free parameters is implemented via a series of wave plates, which perform $2\times 2$ unitaries, and beam displacers, which realize mode permutations. The training of the unitary transformation is performed through the rotation of the motorized half- and quarter-wave plates, using a gradient descent routine. The cost function is evaluated based on the measured photon occupation probability of the upper output mode, which contains only one polarization.

Figure 3

Training starting from different initializations of the unitary. Here, we used two fixed, randomly selected training states to perform the optimization routine. The cost function, which is the occupation probability of the junk mode averaged over the two training states, is plotted against the number of cost function evaluations. The optimization was carried out 20 times, each time starting with a different randomly initialized unitary. To save training time, a training run was terminated once the average occupation probability reached a threshold of 0.02, and the average occupation probability for that training run was thereafter set to the last measured value. The red line shows the mean values of the 20 training runs, with the gray shaded area indicating $\pm 1$ standard deviation of the results. We observe an average occupation probability of $0.03\pm 0.03$ after 160 cost function evaluations. The green circles illustrate a sample trajectory. The uncertainties of individual occupation probability measurements, calculated based on Poissonian counting statistics, are smaller than the markers used.

Figure 4

The effect of the number of training states. The optimization routine was run using one, two, and three training states taken from a compressible subspace. After each training process, 20 random states from the same subspace were prepared and sent through the device to test the compression performance of the unitary. The occupation probability of the junk mode after using one training state (blue circles) was $0.4\pm 0.3$, with two training states (red squares) it was $0.03\pm 0.02$, and with three training states (green diamonds) it was $0.02\pm 0.02$. The uncertainties of individual data points, calculated based on Poissonian counting statistics, are smaller than the markers used.

Figure 5

Test of a channel drift during training. A set of two training states was used to optimize the device under normal conditions, providing a control dataset (red circles) of the occupation probability of the junk mode, averaged over the two training states, versus the number of cost function evaluations. For the green and blue datasets, the same unitary initialization was used, but the necessary encoding was perturbed. This was done by keeping the initial qubits in the state preparation fixed but rotating the wave plate of the scrambling stage by 4° per five cost function evaluations. The datasets of blue squares and green diamonds correspond to different rotation directions. The connecting lines in the plot serve merely as a guide to the eye. The uncertainties of individual occupation probability measurements, calculated based on Poissonian counting statistics, are smaller than the markers used.