In addressing the challenge of Crystal Structure Prediction (CSP), symmetry-aware deep learning models, particularly diffusion models, have been extensively studied, which treat CSP as a conditional generation task. However, ensuring permutation, rotation, and periodic translation equivariance during diffusion process remains incompletely addressed. In this work, we propose EquiCSP, a novel equivariant diffusion-based generative model. We not only address the overlooked issue of lattice permutation equivariance in existing models, but also develop a unique noising algorithm that rigorously maintains periodic translation equivariance throughout both training and inference processes. Our experiments indicate that EquiCSP significantly surpasses existing models in terms of generating accurate structures and demonstrates faster convergence during the training process.

Crystal Structure Prediction (CSP) is crucial in various scientific disciplines. While CSP can be addressed by employing currently-prevailing generative models (**e.g.** diffusion models), this task encounters unique challenges owing to the symmetric geometry of crystal structures---the invariance of translation, rotation, and periodicity. To incorporate the above symmetries, this paper proposes DiffCSP, a novel diffusion model to learn the structure distribution from stable crystals. To be specific, DiffCSP jointly generates the lattice and atom coordinates for each crystal by employing a periodic-E(3)-equivariant denoising model, to better model the crystal geometry. Notably, different from related equivariant generative approaches, DiffCSP leverages fractional coordinates other than Cartesian coordinates to represent crystals, remarkably promoting the diffusion and the generation process of atom positions. Extensive experiments verify that our DiffCSP remarkably outperforms existing CSP methods, with a much lower computation cost in contrast to DFT-based methods. Moreover, the superiority of DiffCSP is still observed when it is extended for ab initio crystal generation.