Quantum Generative Adversarial Autoencoder (QGAA)

• Quantum Generative Adversarial Autoencoder (QGAA) is a hybrid framework that combines a quantum autoencoder’s compression with a GAN’s adversarial sampling of latent spaces.
• The architecture employs variational quantum circuits to encode high-dimensional states into a lower-dimensional latent space and then accurately reconstruct them.
• QGAA has been demonstrated in generating entangled states and synthesizing molecular ground states with chemical-accuracy energy errors on NISQ devices.

A Quantum Generative Adversarial Autoencoder (QGAA) is a hybrid quantum machine learning architecture that integrates a quantum autoencoder with a quantum generative adversarial network, enabling quantum data generation through adversarial learning in a compressed latent space. The QGAA employs a quantum autoencoder to compress high-dimensional quantum states into a low-dimensional latent subspace and then equips this subsystem with generative capability by embedding a QGAN to learn and sample from the latent space. QGAAs are designed for quantum data applications, such as state synthesis and molecular ground state preparation, and have demonstrated chemical-accuracy energy errors in quantum chemistry simulations (Raj et al., 19 Sep 2025).

1. Architecture of the Quantum Generative Adversarial Autoencoder

The QGAA unites two quantum models:

• Quantum Autoencoder (QAE): The QAE employs a variationally parameterized unitary encoder $U_E(\theta_E)$ that maps an %%%%1%%%%-qubit input state from Hilbert space $\mathcal{H}_A$ to a compressed latent subspace $\mathcal{H}_L \otimes \mathcal{H}_T$ ($L<T$), where a subset (“trash” register) is traced out. The decoder $U_D(\phi_D)$ reconstructs the original state using the latent register and a fresh trash register initialized to $|0\rangle^{\otimes n-T}$. The objective is to minimize the average reconstruction infidelity:

$$\mathcal{L}_\mathrm{QAE} = \mathbb{E}_{\sigma_K \sim \{ \sigma_K \} } \left[ 1 - \mathcal{F}(\sigma_K, \rho_K) \right]$$

where $\mathcal{F}$ denotes the state fidelity and $\rho_K$ is the reconstructed state.

• Quantum Generative Adversarial Network (QGAN): The QGAN adversarially trains a generator $U_g(K, \theta_g)$ to reproduce the distribution of latent states $\eta_K$ output by the encoder, conditioned on a label $K$. The discriminator $U_d(K, \theta_d)$ distinguishes between encoder-derived $\eta_K$ (“real”) and generator outputs $\nu_K$ (“fake”). The QGAN’s adversarial objective per label $K$ is:

$$\mathcal{L}_K = \frac{1}{2}\left[ 1 + \mathrm{Tr}(T_K \eta_K) - \mathrm{Tr}(T_K \nu_K) \right]$$

The full adversarial cost is averaged over a training label set $\Lambda_\mathrm{train}$. At the Nash equilibrium, the generator produces latent states indistinguishable by the discriminator: $\nu_K(\theta_g^*) = \eta_K$.

Table: Core QGAA circuit components

Component	Quantum Operation	Subsystem(s)
Encoder ($U_E$)	Parametrized unitary	Input $\to$ Latent, Trash
Decoder ($U_D$)	Parametrized unitary	Latent + $|0\rangle$ Trash $\to$ Output
Generator ($U_g$)	Parametrized quantum circuit	Classical label $\to$ Latent subspace
Discriminator ($U_d$)	Parametrized quantum circuit	Latent subspace

This modular structure imparts the autoencoder with generative capabilities, enabling sampling or conditional generation by first generating latent variables via $U_g$ and decoding them by $U_D$.

2. Training and Optimization Process

Training follows a three-stage pipeline:

1. Autoencoder Pretraining: The QAE is trained using input quantum states $\sigma_K$ to minimize the reconstruction infidelity over a representative data set. After training, the encoder-decoder pair $(U_E^*, U_D^*)$ is fixed.
2. Adversarial Latent Space Learning: With the QAE encoder fixed, a QGAN is trained on the latent subspace $\mathcal{H}_L$. The generator $U_g$ learns to sample compressed (latent) representations $\nu_K$ conditioned on $K$ such that these match the encoder outputs $\eta_K$. The discriminator $U_d$ is optimized to maximize its ability to distinguish real from generated latent states, using cost functions derived from projective measurements:

$$\min_{\theta_g} \max_{\theta_d} \mathcal{L}_\mathrm{QGAN}$$

where

$$\mathcal{L}_\mathrm{QGAN} = \frac{1}{|\Lambda_\mathrm{train}|} \sum_{K \in \Lambda_\mathrm{train}} \mathcal{L}_K$$

1. Generation Phase: At convergence, sampling a classical label $K$ produces a latent state through $U_g(K, \theta_g^*)$, which is then decoded by $U_D^*$ to generate a new quantum output $\xi_K$. For a well-trained model, $\xi_K$ closely approximates the desired target state $\sigma_K$.

3. Generative Quantum State Synthesis

The generative capabilities of QGAA are demonstrated via two tasks (Raj et al., 19 Sep 2025):

• Pure entangled state synthesis: For a family of 2-qubit entangled states

$$|\psi_K\rangle = \cos(k_0/2)|00\rangle + e^{i k_1} \sin(k_0/2)|11\rangle$$

the QAE compresses each $|\psi_K\rangle$ to a 1-qubit latent state. The QGAN is trained to generate this compressed latent state, which is then decoded to recover the original entangled state.

• Quantum chemistry ground state generation: Parameterized molecular ground states for H$_2$ ($n=4$ qubits, compressed to $1$-qubit latent) and LiH ($n=6$, compressed to $4$ qubits) are synthesized. After end-to-end training, the QGAA generates physically meaningful quantum states. Reported mean energy errors are 0.02 Ha (H$_2$) and 0.06 Ha (LiH), which is within the chemical accuracy regime.

Key formula for overlap fidelity in quantum chemistry benchmarking:

$$\mathcal{F}(\sigma_K, \xi_K) = \mathrm{Tr}\left(\sqrt{\sqrt{\sigma_K} \xi_K \sqrt{\sigma_K}}\right)$$

4. Technical Construction and Gradient Evaluation

• Quantum Circuit Ansatz: Both encoder and decoder employ universal parameterized circuits (combinations of single-qubit rotations such as $R_X$, $R_Z$ and two-qubit entangling gates, e.g., $ZZ$ rotations (Dallaire-Demers et al., 2018)), which can approximate arbitrary unitaries given sufficient depth. The generator and discriminator employ similar variational circuit architectures.
• Gradient Measurement: Training exploits quantum-compatible gradient estimation rules. The gradient of an observable $P$ with respect to a circuit parameter $\theta_j$ is given by the commutator-based formula:

$$\frac{\partial}{\partial \theta_j}\langle P(\theta) \rangle = -\frac{i}{2} \mathrm{Tr} \left( \rho_0 U_{1:j}^\dagger \left[ U_{j+1:N}^\dagger P U_{N:j+1}, h_j \right] U_{j:1} \right)$$

These gradients are experimentally accessible using additional ancilla qubits and controlled operations (e.g., via the Hadamard test), and are central for efficiently training both autoencoder and adversarial components on quantum hardware (Dallaire-Demers et al., 2018, Huang et al., 2020).

5. Interpretations, Applications, and Limitations

QGAA enables learning a generative mapping from classical labels (or other input variables) to compressed representations of quantum data, followed by efficient reconstruction. Its hybrid structure is particularly advantageous for:

• Quantum state preparation and sampling: The ability to generate entangled or molecular ground states for varying parameter values demonstrates its potential utility in quantum simulation and state engineering.
• Quantum chemistry and condensed matter: The QGAA's accurate ground state synthesis for small molecules provides warm starts for variational quantum eigensolver (VQE) routines and other quantum chemistry pipelines.
• Near-term quantum machine learning: By using a compressed latent space (requiring fewer qubits), QGAA is well-suited to resource-constrained, noisy intermediate-scale quantum (NISQ) devices.
• General quantum data generation: Any setting where compressed representation and generative modeling of quantum data is required, QGAA offers a scalable, resource-efficient method to synthesize target quantum states.

Limitations: The QAE alone is not generative; its output is restricted to compression and reconstruction. Adding the QGAN component overcomes this, but at the cost of increased circuit complexity and training resource consumption. Accurate adversarial training demands reliable quantum gradient estimation and may be sensitive to hardware noise.

6. Connections to Related Quantum and Hybrid Models

The QGAA concept aligns with broader trends in quantum generative modeling:

• Hybrid Quantum-Classical Adversarial Autoencoders: Prior works employ classical encoders/decoders with quantum latent space sampling (e.g., via quantum Boltzmann machines or quantum annealers (Vinci et al., 2019, Wilson et al., 18 Jul 2024)), but do not offer an end-to-end quantum autoencoder trained with adversarial feedback as in QGAA.
• Hybrid models for classical data generation: Models such as LatentQGAN and VAE-QWGAN combine classical convolutional autoencoders or variational autoencoders with quantum generators, typically for image synthesis (Vieloszynski et al., 22 Sep 2024, Thomas et al., 16 Sep 2024). In contrast, QGAA is designed for quantum data in both input and output.
• Design considerations for hardware implementations: Practical QGAA circuits exploit real-amplitude variational ansatz, parameter-efficient gates, and gradient measurement schemes compatible with superconducting and trapped-ion devices (Nguemto et al., 2022, Raj et al., 19 Sep 2025).

7. Prospects and Open Problems

The QGAA architecture establishes a template for scalable, resource-efficient quantum data generation via adversarial training. Key open problems include:

• Scaling to larger quantum systems: Extending QGAA to higher-dimensional latent spaces and more complex molecular or many-body problems will challenge both classical pre/post-processing and quantum resource efficiency.
• Optimization under noise: Robust training in the presence of quantum hardware noise and limited measurement precision remains an open engineering problem.
• Integration with other quantum models: Extensions to hybrid frameworks leveraging variational quantum encoders/decoders or quantum-enhanced energy-based latent sampling (e.g., quantum Boltzmann, annealers), as well as connections to QUBO-based optimization, are actively under investigation.

In summary, the QGAA formalism implements adversarial learning in quantum latent spaces, providing a practical pathway for data-driven quantum state synthesis, with demonstrated accuracy on representative tasks such as entangled state generation and quantum chemistry, and clear relevance for emerging NISQ-era quantum technologies (Raj et al., 19 Sep 2025).