Crystal structures are defined by the periodic arrangement of atoms in 3D space, inherently making them equivariant to SO(3) group. A fundamental requirement for crystal property prediction is that the model's output should remain invariant to arbitrary rotational transformations of the input structure. One promising strategy to achieve this invariance is to align the given crystal structure into a canonical orientation with appropriately computed rotations, or called frames. However, existing work either only considers a global frame or solely relies on more advanced local frames based on atoms' local structure. A global frame is too coarse to capture the local structure heterogeneity of the crystal, while local frames may inadvertently disrupt crystal symmetry, limiting their expressivity. In this work, we revisit the frame design problem for crystalline materials and propose a novel approach to construct expressive Symmetry-Preserving Frames, dubbed as SPFrame, for modeling crystal structures.

Without zero-mean constraint, the training becomes unstable. Following the training setting of [23], the classifier network is trained with SGD with a weight decay 5e-4, an initial learning rate of 1e-1 and a mini-batch size of 100 for all methods. We use the cosine learning rate decay schedule [49] for a total of 80 epochs. We set the outer level learning ηω as 14 Figure 7: Training curve without zero-mean constraint on CIFAR10 under 40% uniform noise. The MLP weighting network is trained with Adam [51] with a fixed learning rate 1e-3 and a weight decay 1e-4.

A.1 Mach-Zehnder Interferometers (MZIs) A basic coherent optical component used in this work is an MZI. One of the most general MZI structures is shown in Figure 15, consisting of two 50-by-50 optical directional couplers and four phase shifters θ, θ, ω, and ω. An MZI can achieve arbitrary 2 2 unitary matrices SU(2). Figure 15: 2-by-2 MZI with top (T), left (L), upper (P), and lower (W) phase shifters. A.2 MZI-based Photonic Tensor Core Architecture By cascading N(N 1)/2MZIs into a triangular mesh (Recks-style) or rectangular mesh (Clementsstyle), we can construct arbitrary N N unitary U(N).