Machine learning (ML) [1] has been a hot topic for researchers for a long time. Early ML works, such as artificial neurons [2], the perceptron [3], support-vector machines [4], recurrent neural networks [5], and convolutional neural networks [6] were developed and applied in all parts of the human society, such as education ^[7][8][9][7,8,9], agriculture ^[10][11][10,11], finance ^[12][13][12,13], and health care ^[14][15][16][14,15,16]. One of the important goals of ML is generative tasks, where ML programs have to generate new data based on some existing data. Through excellent empirical results, generative adversarial networks (GANs) [17] have been found a brilliant candidate to fulfill this task. In a GAN, a generator and a discriminator play an adversarial game against each other. The generator tries to generate new data that resemble some real data in order to fool the discriminator, while the discriminator aims to distinguish the generated data from the real data.

There has been a sharp increase in the number of GAN variants in the past few years [18]. Since their invention, GANs have been widely used in both semi-supervised and unsupervised learning. However, this type of network bears some issues, such as training instability, mode collapse, and non-convergence [19]. Researchers have been trying their best to improve the efficiencies of the networks, to overcome their shortcomings, and to expand their applications.

In recent years, quantum computing [20] has emerged and drawn a lot of attention from researchers. This new paradigm of computing can accomplish difficult tasks that traditional computing cannot solve. With the development of near-term quantum devices, quantum computing can make use of special properties, including superposition and entanglement, to even perform those tasks exponentially faster [21]. For example, quantum computers can search an unsorted dataset of N entries in time O(N−−√), whereas it takes time O(N)for classical computers to do the same task [22]. Thus, researchers seek to combine quantum computing with machine learning to develop quantum ML. The quantum counterparts of ML models, such as quantum reinforcement learning [23], quantum support vector machines [24], and quantum variational autoencoders [25], were in turn invented.

The generative tasks of ML are getting more and more complicated. Therefore, quantum GANs (QuGANs) were also developed. Basically, QuGANs may have different architectures with various components, in either classical or quantum forms. Instead of neural networks, the quantum parts of a QuGAN consist of variational quantum circuits which also depend on a set of parameters [26]. A certain loss or cost function can be formed, and its gradients with respect to the parameters are calculated for updating the parameters themselves. When an optimal status is reached, the network is able to generate a specific type of data that mimics the true data. The performance of the network can be evaluated using some metrics measuring the distances between these generated data and the real data ^{[27][28][29][30]}[27,28,29,30].

1.1. Generative Adversarial Networks (GANs)

The idea of quantum GANs originates from classical GANs, which were first proposed by Goodfellow et al. [17]. A GAN aims at learning a target distribution, which can be a series of text or a collection of images or audio, and generating samples that have similar characteristics to the samples in the training distribution. The architecture of a typical GAN is depicted in Figure 1. A GAN typically consists of a generator G and a discriminator D. The generator and the discriminator are made from neural networks and are parameterized by θG and θD, respectively. The mission of G is to take noise vectors z from a noise source and produce data x′ that mimic the realistic data x as much as possible in order to fool D, which is supposed to distinguish whether a sample is taken from the real distribution or is generated by G. The distinguishing results y of D are then used to train both G and D iteratively; in other words, θG and θD are updated alternately until the network reaches its optimality. Ideally, this optimality, or Nash equilibrium [17], occurs when G can generate identical samples to those in the real distribution, and D is not able to determine which source the data belong to. GANs are the most potential and popular types of networks used for generative tasks. They vary in terms of network architectures, optimization strategies, and purposes of use. Some notable variants of GANs include deep convolutional GAN ^[31][32], conditional GAN ^[32][33], Wasserstein GAN ^[33][34], boundary equilibrium GAN ^[34][35], Big GAN ^[35][36], Laplacian GAN ^[36][37], and information maximizing GAN ^[37][38]. Many GANs have been the basis of outstanding achievements. For example, StyleGAN3 ^[38][39] obtained a Fréchet inception distance of 3.07 on the FFHQ dataset, 4.40 on the AFHQv2 dataset, and 4.57 on the Beaches dataset. These networks have found their roles in image domains such as image synthesis ^{[35][36][39][40]}[36,37,40,41], image inpainting ^[41][42][42,43], image blending ^[43][44][44,45], image superresolution ^[45][46][46,47], and image-to-image translation ^[47][48][49][48,49,50]; audio domains such as speech synthesis ^[50][51][51,52] and music composing ^[52][53][53,54]; and other fields such as autonomous driving ^[54][55], weather forecasting ^[55][56][56,57], and data augmentation ^[57][58][58,59]. GANs’ effectiveness and varied applications have been described in previous surveys of them ^[18][19][59][18,19,60].

1.2. Quantum–Classical Interface

As quantum machines only work with data stored in quantum states, classical data must be encoded into this form of data. Therefore, data encoding (or embedding) has also become an interesting field of research. There are various data encoding techniques that are used in quantum machine learning algorithms, but in quantum GANs, basis encoding, amplitude encoding, and angle encoding are the most popular ^[60][70]. Basis encoding is the simplest way to embed classical data into a quantum state. The inputs must be in binary form, and each of them corresponds with a computational basis of the qubit system. 

23. Structures of QuGAN

A QuGAN can be fully quantum, hybrid, or based on tensor networks. A diagram of quantum GAN categorization in terms of network architecture is sketched in Figure 2. In the fully quantum (or quantum-quantum) case, both the generator and the discriminator are quantum. Suppose the quantum generator produces fake data with a density matrix ρ, and the quantum discriminator must discriminate those fake data with true data described by a density matrix σ with a measurement operator D. The outcomes of D can be T (i.e., true, or the data are from the real data source), or F (i.e., false, or the data are generated by the generator). Since T and F are positive operators with a 1-norm less than or equal to 1, the set of them is convex, which means there must exist a minimum error measurement [26]. To reach this optimal measurement, the discriminator aims to maximize the probability that it correctly categorizes the data as real or fake by following the gradients of the probability with respect to its parameters to adjust its weights. After that, due to the fact that the set of ρ is convex, the generator also manages to adjust its own weights to maximize the probability that the discriminator fails to discriminate the data and produce an optimal density matrix ρ [26]. Similarly to the traditional GANs, after a number of iterations of adjusting the weights of both the generator and the discriminator, a quantum GAN can also approach the Nash equilibrium. The mechanism is similar when only a part of the classical GAN is replaced by a quantum engine. However, not all hybrid structures are possible. In particular, the generator or the discriminator cannot be classical if the training dataset is generated by a quantum system, which has quantum supremacy. This means the classical generator can never generate the statistics of a dataset that are similar to those of a quantum data source [20]. Therefore, using the same strategy as in a fully quantum network, the quantum discriminator can always find a measurement to distinguish the true and the generated data. As a result, the probability that the discriminator makes wrong predictions will always be less than 12, and the Nash equilibrium will never happen [26]. On the contrary, in the case where the discriminator is classical, both the real data and the generated data which are fed into the discriminator are quantum. When the target data are classical, either the generator or the discriminator can be classical. However, although possessing quantum supremacy, the quantum systems only act on data as quantum states and are unable to work with classical data directly. If the discriminator is quantum, both the training data and the fake data generated by the classical generator need to be encoded before being fed into the discriminator. On the contrary, if the generator is quantum, the generated data must be measured using some computational bases to produce classical generated samples ^[29][30][29,30].

34. Optimization of QuGAN

3.1. Loss Function

4.1. Loss Function

Just like the traditional GANs, the quantum version needs a function whose gradients it can trace along to adjust its weight so as to reach the Nash equilibrium. This function may involve different quantities. Some QuGANs make use of the quantum states of the generator and the discriminator, and some others involve a certain function estimated by the discriminator (in this case, it is called the critic). However, in most QuGANs, the discriminating results of the discriminator are used to determine the loss.

3.2. Gradient Computation

4.2. Gradient Computation

In quantum GANs, there are parametrized quantum circuits that build one or more parts of the network. That part can be the generator, or the discriminator, or both, or even the supplement components to assist the network in working more efficiently. These variational circuits consist of quantum gates or unitaries with tunable continuous parameters. Similarly to the classical counterparts, during the training process, the parameters are updated using a certain optimizing algorithm, such as gradient descent ^[61][88], Adam ^[62][89], or Adagrad ^[63][90]. All these approaches to adjusting the circuits to optimality require calculating the gradients of the cost function, including the partial derivatives with respect to the parameters of the circuits.

3.3. Optimization and Evaluation Strategies

4.3. Optimization and Evaluation Strategies

A good strategy to optimize a quantum GAN is also an important research direction with a lot of interesting questions. For example, what is the order of training the generator and the discriminator that results in the most efficient optimization process? How much should one train the generator in comparison with the discriminator? How many steps of gradients should the network go through? How can the parameters of a network be initialized for the best performance? Additionally, how can one set or even adjust the learning rate to adapt to the change during the training process? These problems matter when training a classical GAN, and they are certainly carried through to the quantum counterparts. With the exponential computational power and the more complicated algorithms, researchers should carefully set up the hyperparameters and determine the optimization strategy before conducting the training process. The solutions to these issues, however, are rather indiscriminate, and due to the different structures of the networks and the elusiveness of quantum nature, there has been no rule to find out the best training strategy that can be applied for every quantum network in general, and for every quantum GAN in particular. To tackle the learning rate and the number of measurement shots, Huang et al. [29] simply performed a grid search to find the optimal hyperparameters. In the same manner, some researchers also set the hyperparameters and adjust them manually during the training process.

45. Quantum GAN Variants

4.1. Fully Quantum GANs

5.1. Fully Quantum GANs

With fully quantum GANs, both generators and discriminators are constructed by quantum circuits, connected directly with the other, and they together apply to a system of qubits ^{[27][28][29][64][65][66][67]}[27,28,29,79,86,87,93]. The target distributions can be quantum, which can be fed directly into the network, or classical, which must be encoded to some quantum states before being input into the network. The workflow of fully quantum GANs is depicted in Figure 34. The noise from latent space puts the quantum system in the state |z⟩. After being applied by the generator, the system is in state |ψ⟩. At this time, in the case the real data are used, the real quantum data source outputs state |x⟩ from the quantum system. The discriminator is then applied, and it changes the state to |y⟩. The states |y⟩ in the cases where the discriminator’s input is real or generated will then be used for the cost function and updating the parameters in the circuits.