Generation of High-Resolution Handwritten Digits with an Ion-Trap Quantum Computer

Generating high-quality data (e.g., images or video) is one of the most exciting and challenging frontiers in unsupervised machine learning. Utilizing quantum computers in such tasks to potentially enhance conventional machine-learning algorithms has emerged as a promising application but poses big challenges due to the limited number of qubits and the level of gate noise in available devices. In this work, we provide the first practical and experimental implementation of a quantum-classical generative algorithm capable of generating high-resolution images of handwritten digits with state-of-the-art gate-based quantum computers. In our quantum-assisted machine-learning framework, we implement a quantum-circuit-based generative model to learn and sample the prior distribution of a generative adversarial network. We introduce a multibasis technique which leverages the unique possibility of measuring quantum states in different bases, hence enhancing the expressivity of the prior distribution. We train this hybrid algorithm on an ion-trap device based on ${}^{171}{\mathrm{Yb}}^{+}$ ion qubits to generate high-quality images and quantitatively outperform comparable classical generative adversarial networks trained on the popular MNIST dataset for handwritten digits.

Popular Summary

Generative modeling is a subdomain of machine learning where the goal is to learn from collected data and generate similar but novel data. In the last decade, we have encountered classical techniques for generative modeling with an impressive performance in application domains ranging from image generation to natural language processing. And although modern computational hardware allows for training large models with billions of parameters, there is still a distinct yearning for a class of powerful, efficient, and compact generative algorithms. Quantum computers promise to fill that gap with their natural ability to encode and access complex data distributions. Our work demonstrates the first large-scale generative algorithm with a quantum component on quantum hardware capable of generating high-resolution images.

In our quantum-assisted approach, we leverage a family of classical algorithms named generative adversarial networks (GANs). While GANs are famous for generating images of fictitious human faces indistinguishable from real humans, they are notorious for displaying learning instabilities. We aim to stabilize and enhance GANs by providing them with quantum measurements from a flexible quantum circuit component. Our framework outperforms comparable conventional GANs on a canonical dataset of handwritten digits and maintains its performance when all quantum circuits are performed on an ion-trap quantum computer.

Our results indicate that current quantum computers can already be applied and tested for practical generative learning tasks using our quantum-assisted framework. As quantum algorithms improve and quantum computers become more robust, we expect quantum models to become a critical component of many large-scale generative algorithms.

Figure 1

Top: schematic description of our QC-AAN framework where the prior of a GAN is modeled by a multibasis QCBM. This quantum generative model with encoded distribution $q(z)$ is trained on activations $z$ in the latent space of the discriminator, learning the feature distribution $p_l$. In the multibasis QCBM, trainable single-qubit rotations follow the parametrized quantum circuit $U(\mathit{\theta })$ and allow for measuring the prepared wave function in additional bases. The angles for these postrotations can be fixed, e.g., at $\pi /2$ to measure $s_o$ with all qubits in the orthogonal $Y$ basis, or they can be trained along with other parameters in $U(\mathit{\theta })$ to measure $s_t$. Measurements in computational basis $s$ and $s_{o/t}$ are concatenated and forwarded as prior samples into the generator network. The GAN is otherwise trained conventionally. Bottom: illustration of the 11-qubit ion-trap quantum device by IonQ based on ${}^{171}{\mathrm{Yb}}^{+}$ ion qubits. The experimental implementation of the QC-AAN algorithm in this work is performed on eight qubits. The device is operated with automated loading of a linear chain of ions, which is then optically initialized with high fidelity. Computations are performed using a mode-locked 355-nm laser, which drives native single-qubit and two-qubit gates.

Figure 2

Digits generated by our QC-AANs with eight simulated qubits and comparable classical DCGANs. The differences in performance are only due to replacing the uninformed random prior of the DCGANs with a multibasis QCBM. All neural-network architectures are equivalent. The models shown here are selected among several training repetitions to have a high IS for their respective model type. The multibasis technique in the ${\mathrm{QC}}_{+o/t}$-AAN enhances the algorithm compared to the 8- and 16-bit DCGAN, and the “plain vanilla” QC-AAN (i.e., without measurements in a second basis).

Figure 3

Left: quantitative comparison between DCGANs with 16-bit random prior distribution and our 8-qubit ${\mathrm{QC}}_{+o/t}$-AAN algorithm. The experimental realization on the IonQ device includes complete implementation of the multibasis QCBM on hardware. Where present, error bars indicate the standard deviation of ten independent training repetitions. The 8-qubit hybrid models generally outperform the classical DCGAN with uninformed prior distribution while the neural-network architectures are equivalent. Right: images of handwritten digits generated by the experimental implementation of the ${\mathrm{QC}}_{+o/t}$-AAN models on the IonQ device. The ${\mathrm{QC}}_{+o}$-AAN model achieves a maximal IS of over 9.4, while the ${\mathrm{QC}}_{+t}$-AAN scores over 9.5 with overall better diversity.

Figure 4

Quantum-circuit ansatz for the QCBM. The ansatz is structured in layers to control expressivity of the model and fidelity of the prepared state. Throughout this work, we use a layer depth of 2 in order to maximize fidelity in the experimental implementation and enforce some interpolation between learned samples. The red numbers indicate the layer-counting convention. For simulations, we use the all-to-all entangling layer as shown here, whereas for the experimental implementation, we instead adopt linear nearest-neighbor connectivity.

Figure 5

Simulation results of QC-AANs with 6 and 8 qubits relative to comparable DCGANs with uniform prior distribution. Depicted are average ISs and standard deviations of ten independent training repetitions per model. We observe no significant improvement for the 6- and 8-qubit QC-AANs, as well as for the ${\mathrm{QC}}_{+o/t}$-AANs with 6 qubits. The first improvements are observed for the 8-qubit ${\mathrm{QC}}_{+o/t}$-AAN, for which we refer to Fig. 3 in the main text.

Figure 6

Schematic neural-network architectures of the generator $G$ and discriminator $D$ used throughout this work for the MNIST dataset of handwritten digits. Purple color indicates a 1D layer of nodes, whereas the orange blocks represent 2D convolutional layers with 128, 64, and 32 channels. The dark color change inside each block indicates the application of a nonlinear activation function after every linear transformation. $G$ and $D$ are approximately inverse, although this is not strictly required. Note that the second to last layer in $D$ represents the latent space which contains $n$ bits, the same size chosen for the prior of $G$.

Figure 7

Example comparison of two hyperparameter configurations for an AAN and QC-AAN, which implement a RBM and QCBM as the model prior, respectively. A training protocol of five training steps per training instance appears to be too much variation in the GAN prior for stable training, but unlike a QCBM, a RBM is prone to becoming unstable under suboptimal configurations and sampling only one distribution mode. When that happens, training of the algorithm fails.

Figure 8

Sketch for two potential scenarios for practical quantum advantage in generative modeling. The large overlapping circles indicate the set of probability distributions which could be expressed within two given parametrized classical or quantum generative models. Note that this does not represent all distributions which could possibly be expressed by any classical or quantum model. (a) The target distribution lies outside the space that can be expressed by the classical model, similar to Fig. 5 in Ref. [13]. This might be due to the distribution not being tractable classically or because this specific parametrized model cannot reach it. (b) The target distribution lies in a region which both models can reach. However, both models do not traverse the space equally, and the classical model might be hindered, for example, by the presence of more local minima or due to pitfalls in conventional heuristics. As indicated in Fig. 7, training success could be significantly different for the classical and quantum algorithms.