Experimental Quantum Generative Adversarial Networks for Image Generation

Quantum machine learning is expected to be one of the first practical applications of near-term quantum devices. Pioneer theoretical works suggest that quantum generative adversarial networks (GANs) may exhibit a potential exponential advantage over classical GANs, thus attracting widespread attention. However, it remains elusive whether quantum GANs implemented on near-term quantum devices can actually solve real-world learning tasks. Here, we devise a flexible quantum GAN scheme to narrow this knowledge gap. In principle, this scheme has the ability to complete image generation with high-dimensional features and could harness quantum superposition to train multiple examples in parallel. We experimentally achieve the learning and generating of real-world handwritten digit images on a superconducting quantum processor. Moreover, we utilize a gray-scale bar dataset to exhibit competitive performance between quantum GANs and the classical GANs based on multilayer perceptron and convolutional neural network architectures, respectively, benchmarked by the Fréchet distance score. Our work provides guidance for developing advanced quantum generative models on near-term quantum devices and opens up an avenue for exploring quantum advantages in various GAN-related learning tasks.

Figure 1

The resource-efficient quantum GAN scheme. (a) The proposed quantum GAN scheme contains a quantum generator $G$ and a discriminator $D$, which can either be classical or quantum. The mechanism of quantum patch GAN is as follows. First, the latent state $|\mathbf{z}⟩$ sampled from the latent space is input into quantum generator $G$ formed by $T$ subgenerators (highlighted in the pink region), where each $G_t$ is built by a PQC $U_{G_t}({\mathit{\theta }}_t)$. Next, the generated image is acquired by measuring the generated states $\{U_{G_t}({\mathit{\theta }}_t)|\mathbf{z}⟩{\}}_{t=1}^T$ along the computation basis. Subsequently, the patched generated image and the real image are input into the classical discriminator $D$ (highlighted in the pink region) in sequence. Finally, a classical optimizer uses the classified results as the output of $D$ to update trainable parameters for $G$ and $D$. This completes one iteration. The mechanism of quantum batch GAN is almost identical to the quantum patch GAN, except for three modifications: (1) we set $T=1$ and introduce the quantum index register into $G$ (highlighted in blue region); (2) the generated state $U_G(\mathit{\theta })|\mathbf{z}⟩$ directly operates with quantum discriminator $D$ implemented by PQC (highlighted in the blue region), where the output is acquired by a simple measurement; and (3) the real image is encoded into the quantum state to operate with $D$. (b) The implementation of PQC used in the quantum generator and quantum discriminator. (c) The machinery of quantum generators employed in quantum patch and batch GANs. For quantum batch GAN, an index register with extra operations should be involved when the batch size is larger than one. (d) The quantum discriminator employed in the quantum batch GAN. To attain nonlinear property, two generated states are fed into the quantum discriminator simultaneously.

Figure 2

Handwritten digit image generation. (a),(b) Experimental results for the handwritten digit “0” and “1,” respectively. From top to bottom, the first row illustrates real data examples, the second and third rows show the examples generated by quantum patch GAN trained using a superconducting processor and noiseless numerical simulator, respectively. The number of parameters for quantum generator is set to 100, and the total number of iterations is about 350.

Figure 3

Gray-scale bar image generation. (a) Experiment results of quantum patch GAN for a $2\times 2$ image dataset. The left panel illustrates real examples and generated examples. The right panel, highlighted in the yellow region, shows boxplots that illustrate the FD scores achieved by different generative models. A lower FD score equates to better performance of the generative model. Specifically, we set the number of trainable parameters for $G$ as $N_p=9$, the number of iterations as 350, sample 1000 generated examples to evaluate the FD score after every 50 iterations, and repeat each setting 5 times to collect statistical results. The label “$\mathrm{para}\mathrm{\_}\ell$” refers to the FD score of the classical GAN employing the generator with $\ell$ trainable parameter. The labels “Q_exp” and “Q_sim” refers to FD scores of quantum GANs built using a quantum processor and noiseless numerical simulator, respectively. The labels “Min_Clc” and “Min_Exp” represents the achieved best FD scores for classical and quantum GANs, respectively. The left FD score plot compares the performance between classical and quantum GAN where $\ell$ approximates to $N_p$, i.e. $\ell =10$, and the results show that quantum GAN have a better performance than GAN MLP and GAN CNN with a similar number of parameters. The middle and right FD score plots show the required value $\ell$, i.e. $\ell =18(18)$ and $\ell =60(57)$ for GAN MLP (GAN CNN), which enables the classical GANs to achieve the comparable and even better performance over quantum GANs. The performance is evaluated by the average score (middle line of the shaded box) and the minimal FD score. (b) Experiment results of quantum batch GAN for the $2\times 2$ image dataset. The three plots indicate that the quantum batch GAN could achieve a similar performance to the quantum patch GAN.

Figure 4

An example of FCNN. The left panel illustrates the basic structure of FCNN that consists of an input layer, one hidden layer, and an output layer. In the green region, the number of neurons for the input layer and the first hidden layer is 2 and 5, respectively. The right panel shows the calculation rule of a single neuron. The neuron, highlighted by the gray region, is calculated by $f(⟨\mathit{\theta },\mathbf{x}⟩)$, where $\mathit{\theta }$ represents the weight, $\mathbf{x}$ refers to the outputs of the green neurons, and $f(\cdot )$ is the predefined activation function.

Figure 5

An example of the boxplot. The gray circle refers to the outlier of the given data. The orange line represents the median of the given data. The two thick gray lines correspond to the upper extreme and lower extreme, respectively. The upper edge and lower edge of the gray box stand for the upper hinge and the lower hinge, respectively. The distance from the upper extreme to the upper hinge (or from the lower extreme to the lower hinge) equals to $1.5\mathrm{IQR}$.

Figure 6

The implementation of MPQC. (a) A general framework of MPQC. The trainable unitary $U_l(\mathit{\theta })$ with $l\in [L]$ refers to the $l\mathrm{th}$ layer of MPQC. The arrangement of quantum gates in each layer is identical. (b) A paradigm for the trainable unitary $U_l(\theta )$ and $U_E$. For $U_l(\theta )$, the trainable qubit gates $U_S$ are rotation single-qubit gates along the $Z$ and $Y$ axis. The trainable parameter refers to the rotation angle. For $U_E$, the fixed two qubits gates, i.e., cz gates, are applied onto the adjacent qubits.

Figure 7

Quantum GAN. (a) The implementation of $G_t$ for $t$ quantum patch GAN. The subgenerator $G_t$, or equivalently, the trainable unitary $U_G({\mathit{\theta }}_t)$, is constructed by PQC and highlighted by the blue region with a dashed outline. Let $U_G({\mathit{\theta }}_t):=\prod _{l=1}^L[U_E{U}_l({\mathit{\theta }}_t)]$, where $U_l({\mathit{\theta }}_t):=\underset{i=1}{\overset{N}{⨂}}[U_S({\mathit{\theta }}_t^{(i,l)})]$ is the $l\mathrm{th}$ trainable layer, $U_S({\mathit{\theta }}_t^{(i,l)})$ is the trainable unitary with $U_S\in SU(2)$, and $U_E$ is the entanglement layer with $U_E:=\underset{i=1}{\overset{2i+1\le N}{⨂}}\mathrm{cz}(2i,2i+1)\underset{i=1}{\overset{2i\le N}{⨂}}\mathrm{cz}(2i-1,2i)$. For example, we set $U_S(\mathit{\theta })=R_Y(\mathit{\theta })$, $L=3$, and $N=3$ to accomplish the gray-scale bar image generation in case $m=2$, where the employed qubits are highlighted by the blue line and the used quantum gates are highlighted by yellow region. (b) The main architecture of the quantum batch GAN. A pretrained unitary $U({\mathit{\alpha }}_z)$, the quantum generator $U_G({\mathit{\theta }}^{(k)})$, and the quantum discriminator $U_D({\mathit{\gamma }}^{(k)})$ are applied to the input state $|0{⟩}^{\otimes N}$ in sequence. We adopt the same rules used in (a) to build $U_G({\mathit{\theta }}^{(k)})$ and $U_D({\mathit{\gamma }}^{(k)})$.

Figure 8

Experiment setup. There are 12 qubits in total in our superconducting quantum processor, from which we choose six adjacent qubits labeled with $Q_1$ to $Q_6$ to perform the experiment. Each qubit couples to a corresponding resonator for state readout. For each qubit, individual capacitively coupled microwave control lines ($XY$) and inductively coupled bias lines ($Z$) enable full control of qubit operations.

Figure 9

The simulation results for approximating discrete Gaussian distribution with finite measurements $K=3000$. The left panel shows the performance of approximated discrete Gaussian. The label “target” refers to the target Gaussian distribution to be approximated. The label “LR” refers to learning rate. Similarly, the right panel illustrates the corresponding training loss.

Figure 10

Architectures of the employed two types of classical GANs. (a) The generator of GAN MLP, a classical GAN model with multilayer perceptron generator. (b) The generator of GAN CNN, a classical GAN model with convolutional generator.