This work introduces quantum-inspired tensor-network circuits as trainable transforms for image inpainting. Among the proposed architectures, the diagonal quantum Fourier transform (QFT) relaxation is invertible with
O(N2logN) computational cost for
N×N images, inherently preserving minimum coherence throughout training via its circuit structure and eliminating the need for explicit coherence penalties. Unconstrained gradient-based phase optimization (Riemannian-optimization free) enables efficient learning from randomly sampled training data, allowing the learned transform to generalize to test images observed through fixed sampling masks. Numerical tests show that the learned models outperform fixed transforms and per-image optimization while matching the performance of much larger unitary architectures, yet with far fewer parameters.