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

AI summary

Overview

Research area: Quantum machine learning, specifically quantum generative adversarial networks (QGANs) for image generation, combined with ideas from quantum implicit neural representations (QINRs).

Technical level: Advanced. The paper assumes familiarity with parameterized quantum circuits, Pauli measurements, Wasserstein GAN training, and implicit neural representations.

Scope: A simulation-only study (MNIST and Fashion-MNIST at 28×28) proposing CoQui, a QGAN that replaces amplitude-encoded pixel mapping with per-coordinate quantum circuit evaluation, together with a purpose-built circuit and ablations against PQWGAN, an end-to-end "Wasserstein QGAN," and a classical INR-GAN.

What This Paper Is About

Existing QGANs for images map quantum-state amplitudes onto pixel intensities, which forces the state dimension (or the number of address qubits) to grow with image resolution, and forces many pixels to share one normalized quantum state so they compete for probability mass — the authors point to PQWGAN's systematically dimmer outputs and to uneven address-amplitude distributions in Jäger et al.'s model as symptoms. CoQui instead treats an image as a continuous function: a spatial coordinate plus a shared latent code are fed to a classical network that parameterizes a quantum circuit, and the pixel value is read from the measurement of one dedicated "color" qubit. A full image is produced by querying every coordinate.

Key Contributions

1. A new formulation of quantum image generation. CoQui (Coordinate-conditioned Quantum Implicit GAN) casts generation as learning a continuous mapping from spatial coordinates and latent variables to pixel intensities, rather than amplitude-to-pixel mapping. This decouples qubit count from image resolution and removes the shared probability-normalization constraint across pixels.
2. A quantum circuit with structural inductive bias. The generator uses one color qubit plus N_f feature qubits, with scaled data re-uploading, local feature-qubit rotations, ring-topology CNOT entanglement, controlled-R_Y "feature-to-color writing," and residual color-qubit updates, so the color output depends explicitly on the encoded coordinate/latent features.
3. A direct observable-based pixel readout. Pixel intensity is defined as (1 − ⟨Z_0⟩)/2 from the Pauli-Z expectation of the color qubit of a separately evaluated coordinate-conditioned state, so the quantum circuit itself performs pixel-level modeling.
4. A controlled empirical comparison. Simulated experiments on MNIST and Fashion-MNIST against PQWGAN, an end-to-end Wasserstein QGAN, and a classical INR-GAN, plus circuit-architecture, core-component, and capacity ablations.

Main Findings

• Large resource advantage: CoQui uses 5 qubits and 1 quantum circuit, versus 192 qubits and 32 circuits for PQWGAN and 11 qubits and 1 circuit for the Wasserstein QGAN baseline (Table 2). The authors note amplitude-mapping methods scale qubits with resolution, whereas CoQui's qubit count does not.
• Better class-wise FID than quantum baselines: CoQui achieves the best or tied-best FID on four MNIST classes and seven Fashion-MNIST classes relative to the classical INR-GAN of similar parameter scale. Class-wise FID for CoQui ranges from 16 to 26 on MNIST (e.g., 17 for class 0, 26 for class 2, 16 for class 9) and from 28 to 62 on Fashion-MNIST, versus 97 to 275 for PQWGAN and 112 to 246 for the Wasserstein QGAN on Fashion-MNIST.
• Competitive in the harder ten-class setting: In Figure 2's full-dataset annotations, PQWGAN scores FID 221 and 244, while CoQui and the classical INR-GAN produce the lowest scores in each row — 34 and 42 in the first row, and 72 and 72 in the second. The authors state the ten-class setting is harder than class-wise generation and that all methods degrade.
• Proposed circuit beats a hardware-efficient ansatz: FID improves from 41.41 ± 6.10 to 40.15 ± 3.45, JSD from 0.00858 ± 0.00333 to 0.00411 ± 0.00128, entropy from 0.9856 ± 0.0055 to 0.9930 ± 0.0021, P@5 from 0.489 ± 0.081 to 0.505 ± 0.061, and Sobel discrepancy from 0.0200 ± 0.0000 to 0.0073 ± 0.0033; the hardware-efficient version had a slightly higher R@5 (0.485 vs 0.467).

Authors’ abstract

Quantum generative adversarial networks (QGANs) have attracted increasing attention for image generation using parameterized quantum circuits. Existing amplitude-based approaches face two key limitations: pixel locations are typically encoded by computational-basis indices or address qubits, causing quantum resources to grow with image resolution; meanwhile, jointly decoding many pixels from normalized quantum states introduces probability competition among pixels and limits precise pixel-wise control. To address these issues, we reformulate quantum image generation as coordinate-conditioned implicit function learning. Our method takes spatial coordinates and latent variables as inputs, uses a classical embedding network to generate input-dependent circuit parameters, and evaluates a variational quantum circuit at each coordinate. Pixel intensities are directly obtained from the expectation value of a dedicated color qubit, and a complete image is generated by querying all spatial coordinates. This design decouples image resolution from address-qubit requirements and avoids shared probability-normalization constraints across pixels. We further design a specialized variational quantum circuit to provide structural inductive bias for coordinate-conditioned generation. Simulated experiments on two benchmark datasets show that our method outperforms FRQI-based generation and PQWGAN in visual and quantitative quality while using fewer qubits, and also achieves better generation quality than the corresponding classical baseline.