In quantum image processing, a fundamental step is encoding classical image data into quantum states. This can be achieved using methods such as Flexible Representation of Quantum Images (FRQI), Quantum Probability Image Encoding (QPIE), and Novel Enhanced Quantum Representation (NEQR). However, on real quantum hardware, these encodings can quickly lead to circuits with many gates, large circuit depth, and high qubit usage, which is a problem for Noisy Intermediate-Scale Quantum (NISQ) devices. In this work, we investigate whether low-rank state approximation, formulated via Schmidt decomposition, can help reduce this complexity. The method keeps only the most significant parts of a quantum state's entanglement structure, making state preparation more efficient while preserving most of the image information. We compare the three encoding techniques in their original form and with low-rank approximation, evaluating metrics such as circuit depth, CNOT count, MSE, and visual quality of reconstructed images. The results reveal meaningful trade-offs between accuracy and resource efficiency, with the FRQI model achieving a 97 percent reduction in circuit depth while maintaining a near-perfect reconstruction (MSE of about 0.27). This demonstrates the potential of low-rank techniques for advancing practical quantum image processing on near-term hardware.

翻译：在量子图像处理中，基本步骤是将经典图像数据编码为量子态。这可通过柔性量子图像表示（FRQI）、量子概率图像编码（QPIE）和新型增强量子表示（NEQR）等方法实现。然而，在实际量子硬件上，这些编码方式会迅速导致电路包含大量量子门、电路深度大、量子比特使用量高，这对含噪声中等规模量子（NISQ）器件而言是一个问题。本文研究了基于施密特分解的低秩态近似能否帮助降低这种复杂性。该方法仅保留量子态纠缠结构中最显著的部分，使态制备更高效，同时保留大部分图像信息。我们比较了这三种编码技术的原始形式与低秩近似形式，评估了电路深度、CNOT计数、MSE和重建图像的视觉质量等指标。结果显示，在精度与资源效率之间存在有意义的权衡，其中FRQI模型在保持近乎完美的重建（MSE约为0.27）的同时，电路深度减少了97%。这表明低秩技术在推动近期硬件上实用量子图像处理方面具有潜力。