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High-resolution imaging systems have radically developed in areas such as medical diagnostics, remote sensing, and autonomous systems thus creating an urgent need to develop advanced algorithms that can be used to restore an image and segment it at the same time. The classical CNN- and transformer algorithms operate largely in a spatial space and cannot capture global frequency information and non-linear spectral relationships. To address these drawbacks, we propose QFIT-FNO (Quantum Frequency -Integrated Fourier Neural Operator) - a novel, quantum-inspired model with a frequency-aware feature representation capability achieved by combining Fourier spectral learning with quantum-inspired frequency interaction transformations (QFIT). The architecture implies that in the first place, input images are coded in the spectral domain using Fourier Neural Operator to encode long-range interactions and global properties of the architecture. It relies on the principles of quantum amplitude encoding and entanglement and followed by the complicated crossfrequency interactions modeled with QFIT module to enhance the discriminative feature learning. These enhanced spectral representations are majed with spatial-domain representations through spectral-spatial fusion layer, which is offered with supplementary global and local information. Finally, a multi-task decoder is a decoder that simultaneously achieves both image restoration and segmentation, these contribute to the recovery faith and boundary accuracy. Comprehensive benchmarking of QFIT-FNO shows it is able to surpass leading CNN, transformer, and spectral models by wide margins - with up to 1.5 dB PSNR and 2.4 % Dice score. The work introduces a novel paradigm on the frequencysensitive visual understanding and paves the way to the implementation of quantum-inspired spectral models of the next-generation imaging systems.