A quantum color space for finite-shot RGB reconstruction in quantum image processing

Technical Summary: A Quantum Color Space for Finite-Shot RGB Reconstruction

Problem Statement
Quantum image processing (QIP) requires color representations that are both compact within quantum registers and recoverable from finite-shot measurement statistics. While existing Quantum Image Representations (QIRs) like FRQI, MCQI, and NEQR have focused on storage efficiency and indexing, they often treat color as a value to be stored rather than addressing the geometric organization of color information. A critical gap exists in understanding how classical RGB coordinates map to quantum state space and how finite-shot measurement perturbations affect the stability of color reconstruction. Current schemes often lack a clear framework for how measurement-derived estimates are decoded back to classical colors, particularly when the encoded color space is dense and decision boundaries are narrow.

Methodology
The authors propose a Bloch-sphere-based quantum color space that establishes a structure-preserving correspondence between discrete RGB values and single-qubit state parameters $(\theta, \phi)$.

1. Geometric Mapping:

   • RGB triplets are mapped to spherical coordinates on the Bloch sphere.
   • Polar Angle ($\theta$): Organizes colors based on maximum–minimum channel subgroups ($M_x, M_n$). Colors sharing the same $M_x$ and $M_n$ are assigned to common latitude bands.
   • Azimuthal Angle ($\phi$): Encodes channel-permutation trajectories within each latitude band.
   • Grayscale Subspace: The diagonal set $\{(x,x,x)\}$ naturally forms a one-dimensional subspace where intensity is encoded solely by $\theta$, rendering $\phi$ redundant.

2. Quantization and Grids:

   • The study evaluates three quantized RGB grids: $64^3$, $128^3$, and $256^3$. These are not raw intensity ranges but display-scale grids (e.g., $128^3$ corresponds to $\{0, 2, 4, \dots, 254\}^3$).

3. Decoding Strategy:

   • Instead of assuming exact analytic inversion, the method uses a Region-of-Point (RoP) and Lookup-Table (LUT) rule.
   • Finite-shot measurement counts are converted into angle estimates $(\hat{\theta}, \hat{\phi})$.
   • These estimates are decoded into RGB values by determining which precomputed decision region $I(r,g,b)$ the estimated angles fall into.

4. Experimental Protocols:

   • Pixel-wise Validation: Each pixel is processed by an independent two-qubit circuit. This isolates intrinsic finite-shot recoverability.
   • Position-Register Full-Image Encoding: All pixels are encoded in a single circuit using position qubits. The total shot budget is shared across all positions and angle branches, simulating a more compact but measurement-sparse scenario.
   • Datasets: MNIST (for grayscale subspace validation) and CIFAR-10 (for RGB reconstruction).
   • Simulation: Qiskit Aer simulations without explicit hardware noise models, focusing on statistical finite-shot effects.

Key Results

• Grayscale Reconstruction (MNIST): The induced grayscale subspace demonstrated high stability. Reconstruction achieved an average Mean Squared Error (MSE) of 3.90 and a Peak Signal-to-Noise Ratio (PSNR) of 42.44 dB. The one-dimensional nature of the grayscale diagonal ($\phi$ redundancy) contributed to this robustness.
• RGB Reconstruction (CIFAR-10):
  • Quantization-Separability Trade-off: The $64^3$ grid yielded the best performance (MSE: 81.29, PSNR: 29.84 dB, SSIM: 0.9313).
  • Denser Grids: The $128^3$ grid showed moderate degradation, while the full $256^3$ grid performed the worst (MSE: 609.77, PSNR: 20.55 dB, SSIM: 0.7435).
  • Cause: Higher nominal color resolution creates narrower angular decision regions (RoP). Finite-shot angle estimation errors, even if small, are more likely to cross decision boundaries in dense grids, leading to color mis-decoding.
• Shot-Wise Analysis:
  • Increasing the shot budget improves angle estimation accuracy. However, for a fixed residual angular error, coarser grids ($64^3$) remain more robust due to wider decision regions.
  • Measurement-Allocation Bottleneck: In the position-register full-image mode, reconstruction quality degraded significantly across all grids (MSE $\approx$ 890, PSNR $\approx$ 18.7 dB). The shared shot budget resulted in sparse per-position measurements (approx. 128 samples per pixel/branch at 218 total shots), making reliable recovery of both $\theta$ and $\phi$ impossible compared to the pixel-wise mode.

Significance and Claims
The paper argues that quantum color representations must be designed not only for compact storage but specifically for stable and interpretable recovery from finite-shot measurements.

• Geometric Organization: The proposed Bloch-sphere construction explicitly links the geometry of the classical RGB cube to quantum state parameters, making measurement dependence explicit.
• Stability over Resolution: The results challenge the assumption that higher classical color resolution automatically improves quantum reconstruction. Instead, reconstruction quality is governed by the balance between nominal resolution and finite-shot angular separability. Coarser grids provide larger angular margins, enhancing robustness against statistical noise.
• Encoding Trade-offs: The study highlights a practical trade-off between compact image-level encoding (position-register) and reliable per-pixel recovery. While compact encoding saves qubits, it introduces a measurement-allocation bottleneck that severely limits reconstruction fidelity under realistic shot budgets.

The authors conclude that future quantum image processing frameworks should prioritize measurement-stable recovery strategies, such as adaptive color-space partitioning and likelihood-based decoding, rather than focusing solely on qubit efficiency.