⚛️ quantum physics

CoQui: A Coordinate-Conditioned Quantum Implicit Generative Adversarial Network for End-to-End Image Generation

The paper introduces CoQui, a coordinate-conditioned quantum implicit generative adversarial network that overcomes the scalability and control limitations of existing amplitude-based QGANs by generating images through a variational quantum circuit queried at spatial coordinates, thereby decoupling resolution from qubit requirements and achieving superior visual and quantitative performance with fewer resources.

Original authors: Xue Yang, Rigui Zhou, ShiZheng Jia, Dax Enshan Koh, Siong Thye Goh, Young-Wook Cho, YaoChong Li, Xuezhi Ma, Hongyu Chen, Xin Wang

Published 2026-08-13

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Xue Yang, Rigui Zhou, ShiZheng Jia, Dax Enshan Koh, Siong Thye Goh, Young-Wook Cho, YaoChong Li, Xuezhi Ma, Hongyu Chen, Xin Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where computers don't just crunch numbers but dance with the very fabric of reality. This is the realm of quantum computing, a field that uses the strange, counter-intuitive rules of subatomic particles to solve problems that would take traditional supercomputers forever. One of the most exciting applications of this technology is "generative AI," where computers learn to create new things, like art or music, rather than just analyzing what already exists. To do this, scientists use a clever game called a "Generative Adversarial Network" (GAN). Think of it as a forger and a detective: the forger tries to create fake images so perfect that the detective can't tell them apart from real photos. As they play this game over and over, the forger gets incredibly good at making art. Now, imagine giving this forger a quantum superpower. That's the goal of "Quantum GANs"—using quantum computers to generate images. But here's the catch: current quantum computers are tiny and fragile, and the old ways of making them draw pictures are like trying to paint a massive mural using only a single, tiny brush that gets clogged if you try to paint too many pixels at once.

Enter CoQui, a new approach that changes the rules of the game. Instead of forcing the quantum computer to juggle all the pixels of an image at once, CoQui treats the image like a magical, continuous landscape. Imagine you have a magical paintbrush that can instantly tell you the color of any spot on a canvas just by asking, "What color is at coordinate (x, y)?" You don't need to paint the whole picture at once; you just visit every single point, one by one, and ask the brush for its color. This is what CoQui does. It reformulates image generation as a "coordinate-conditioned" task. Instead of mapping the quantum state's "amplitudes" (a technical way of saying the probability of a particle being in a certain state) directly to pixel brightness, CoQui feeds the quantum computer a specific location (coordinates) and a secret code (latent variable). The quantum computer then acts like a sophisticated calculator, spitting out the exact brightness for that specific spot.

The paper suggests that this method solves two major headaches of previous quantum image generators. First, old methods required the number of quantum bits (qubits) to grow as the image got bigger, like needing a bigger and bigger bucket to hold more water. CoQui breaks this link; you can generate a high-resolution image without needing a massive quantum computer. Second, old methods made pixels "fight" for probability mass, meaning if one pixel got too bright, another had to get too dark. CoQui lets each pixel be measured independently, so they don't have to compete. The researchers tested this idea in computer simulations using standard image datasets (MNIST and Fashion-MNIST). They found that CoQui could create clearer, sharper images than previous quantum methods while using significantly fewer resources—specifically, just 5 qubits (one for the color and four for features) and a single quantum circuit, compared to methods that needed 192 qubits or complex patch-by-patch assembly.

The team also designed a special "quantum circuit" architecture that acts like a smart filter, allowing the features of the image to control the color output in a very structured way. In their simulations, this design helped the model learn better and produce images with less blur and more detail. While the results are promising, the authors are careful to note that these are simulations run on classical computers, not tests on actual, noisy quantum hardware. They suggest that this coordinate-based approach offers a more flexible and efficient path forward for quantum image generation, potentially allowing future quantum devices to create complex images without needing millions of qubits. It's a step toward a future where a tiny quantum chip could paint a masterpiece, one pixel at a time.

Technical Summary: CoQui – A Coordinate-Conditioned Quantum Implicit Generative Adversarial Network

Problem Statement

Existing Quantum Generative Adversarial Networks (QGANs) for image generation predominantly rely on a paradigm where pixel intensities are mapped directly from quantum-state amplitudes. The authors identify two critical limitations in this approach:

Scalability Issues: Encoding pixel locations via computational-basis indices or address qubits causes the required quantum-state dimension or the number of qubits to grow linearly with image resolution.
Pixel Coupling and Control: Existing methods jointly decode multiple pixels from one or a few normalized quantum states. This forces pixels to compete for probability mass, making it difficult to control individual pixel values precisely. This often results in generated images with lower overall brightness compared to real data distributions and uneven amplitude distributions.

Furthermore, many existing approaches use quantum circuits only to model compressed classical representations, rather than directly undertaking pixel-level generation.

Methodology: CoQui Framework

To address these limitations, the paper proposes CoQui (Coordinate-conditioned Quantum Implicit GAN), which reformulates image generation as a coordinate-conditioned implicit function learning problem. Instead of mapping amplitudes to pixels, CoQui learns a continuous mapping from spatial coordinates to signal values.

1. Core Architecture

The framework operates as follows:

Inputs: The generator takes spatial coordinates $c = (x, y)$ and a latent variable $z$ as inputs.
Classical Embedding: A classical neural network $\Gamma$ processes the concatenated positional encoding of $c$ and the latent code $z$ . It outputs parameters that define the rotation angles for a Parameterized Quantum Circuit (PQC).
Quantum Generator: The PQC evaluates the image at specific coordinate locations.
- Qubit Allocation: The circuit uses $N_q = N_f + 1$ qubits. One qubit ( $q_0$ ) serves as the color qubit for pixel readout, while the remaining $N_f$ qubits act as feature qubits encoding implicit spatial features.
- Circuit Structure: The circuit consists of $L$ $L$ repeating layers containing:
  - Color-Qubit Brightness Initialization: A trainable $R_Y$ rotation to set an initial average brightness.
  - Scaled Data Re-uploading: Layer-specific scale and bias parameters modulate the input angles, allowing each layer to emphasize or attenuate coordinate-latent signals.
  - Local Rotations & Entanglement: Feature qubits undergo local rotations and ring-topology CNOT entanglement to model correlations.
  - Feature-to-Color Writing: Controlled- $R_Y$ gates transfer information from feature qubits to the color qubit, making the color output explicitly dependent on the feature representation.
  - Color Residual Update: Additional trainable rotations on the color qubit allow for refinement of pixel intensities across layers.
Pixel Readout: The pixel intensity is not derived from state amplitudes but is directly read out as the expectation value of the Pauli-Z observable on the color qubit: $G(c, z) = (1 - \langle Z_0 \rangle)/2$ .
Image Synthesis: A complete image is generated by querying the generator at all spatial coordinates $(i/W, j/H)$ .

2. Training Objective

CoQui is trained using the Wasserstein GAN objective with a gradient penalty (WGAN-GP), employing a classical discriminator (critic) to distinguish between real and generated images.

Key Contributions

The paper outlines three primary contributions:

Coordinate-Conditioned Paradigm: CoQui decouples the number of qubits from image resolution by treating generation as a continuous mapping from coordinates to pixel intensities. This eliminates the inter-pixel competition for probability mass inherent in amplitude-mapping methods.
Structurally Biased Quantum Circuit: The authors design a specialized PQC with structural inductive bias. By using feature qubits to controllably modulate a dedicated color qubit, the model enhances its capability for conditional pixel generation and improves expressiveness.
Empirical Performance: Systematic simulations demonstrate that CoQui achieves superior visual quality and quantitative performance compared to amplitude-mapping baselines (PQWGAN and Jäger et al.'s Quantum Wasserstein QGAN) while utilizing significantly fewer qubits. It also shows competitive performance against a classical baseline of similar parameter scale.

Experimental Results

Experiments were conducted on MNIST and Fashion-MNIST datasets (28x28 resolution) using a simulated environment (JAX framework).

Resource Efficiency: CoQui generates full images using only 5 qubits (1 color + 4 feature) and a single quantum circuit. In contrast, amplitude-mapping baselines require significantly more resources (e.g., PQWGAN requires 192 qubits and 32 circuits for the same task).
Quantitative Performance:
- CoQui achieved the best or tied-best Fréchet Inception Distance (FID) scores on 4 out of 10 MNIST classes and 7 out of 10 Fashion-MNIST classes compared to a classical INR-GAN.
- It outperformed PQWGAN and the standard setting of the Quantum Wasserstein QGAN in both visual quality and FID scores on the full 10-class datasets.
- CoQui demonstrated better low-level visual statistics (brightness and Sobel edge consistency) compared to amplitude-mapping methods, which produced blurrier samples due to pixel coupling.
Ablation Studies:
- Circuit Architecture: The proposed structured circuit (feature-to-color modulation) outperformed a hardware-efficient ansatz in FID, JSD, and edge statistics.
- Components: Removing the scaled data re-uploading caused the most significant performance degradation (FID increased from ~40 to ~56), highlighting the importance of layer-wise input scaling.
- Capacity: Increasing feature qubits beyond 4 or circuit depth beyond 20 layers yielded diminishing returns or slight degradation, suggesting that shallow-to-moderate depth circuits are sufficient for this task.

Significance and Claims

The paper claims that CoQui represents a significant step forward in end-to-end quantum image generation. By shifting from amplitude-mapping to coordinate-conditioned implicit learning, the method addresses the fundamental scalability bottleneck of QGANs, allowing for high-resolution image generation without a proportional increase in qubit count.

The authors emphasize that their approach provides a more flexible route for modeling image distributions, enabling precise control over individual pixel values without the constraints of global probability normalization. While the current evaluation is limited to classical simulations on grayscale benchmarks without hardware noise, the work establishes a foundation for future deployment on real quantum devices, with potential extensions to higher resolutions and color image generation.