This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC’2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves certainty of success, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it impossible to have a deterministic quantum algorithm with exponential speedups for the welded tree problem.

Deep learning has emerged as the predominant approach for multispectral and hyperspectral image fusion. However, most fusion networks are typically trained and validated on hyperspectral and multispectral image pairs generated from the same hyperspectral images, with degradation simulations inconsistent with real situations and relatively limited volumes of images. When transferring a pretrained multispectral and hyperspectral image fusion model from ground or airborne images to spaceborne images, it encounters a larger dataset and more complex spatial-spectral degradation, leading to spectral distortions and spatial artifacts in the fused images. In this article, the challenges associated with the transfer are addressed through the introduction of a self-supervised multispectral and hyperspectral image fusion unrolling network for spaceborne imagery, termed as MH-FUNet. MH-FUNet adopts a self-supervised paradigm to learn a robust mapping from spaceborne data. It utilizes a deep unrolling network to iteratively refine fusion results from coarse to fine. To account for spatial scale differences between the self-supervised training and test datasets, a multiscale fusion strategy is introduced. This strategy is combined with spectral and spatial attention mechanisms to restore spatial and spectral details. Additionally, a gradient constraint unit is proposed to maintain spatial consistency when up-scaling low-resolution hyperspectral imagery. Performance evaluation of the proposed method is conducted against state-of-the-art fusion techniques on both simulated Chikusei dataset and the proposed real WHU-MHF dataset, which consists of simultaneously observed hyperspectral and multispectral image pairs. MH-FUNet outperforms existing methods across all datasets, demonstrating superior performance in spaceborne multispectral and hyperspectral image fusion experiments.

Internal symmetry networks are a recently developed class of cellular neural network inspired by the phenomenon of internal symmetry in quantum physics. Their hidden unit activations are acted on non-trivially by the dihedral group of symmetries of the square. Here, we extend Internal symmetry networks to include recurrent connections, and train them by backpropagation to perform two simple image processing tasks.