We introduce Clifford Group Equivariant Neural Networks: a novel approach for
constructing $\mathrm{E}(n)$-equivariant networks. We identify and study the
$\textit{Clifford group}$, a subgroup inside the Clifford algebra, whose
definition we slightly adjust to achieve several favorable properties.
Primarily, the group's action forms an orthogonal automorphism that extends
beyond the typical vector space to the entire Clifford algebra while respecting
the multivector grading. This leads to several non-equivalent
subrepresentations corresponding to the multivector decomposition. Furthermore,
we prove that the action respects not just the vector space structure of the
Clifford algebra but also its multiplicative structure, i.e., the geometric
product. These findings imply that every polynomial in multivectors, including
their grade projections, constitutes an equivariant map with respect to the
Clifford group, allowing us to parameterize equivariant neural network layers.
Notable advantages are that these layers operate directly on a vector basis and
elegantly generalize to any dimension. We demonstrate, notably from a single
core implementation, state-of-the-art performance on several distinct tasks,
including a three-dimensional $n$-body experiment, a four-dimensional
Lorentz-equivariant high-energy physics experiment, and a five-dimensional
convex hull experiment.

介绍了一种新的方法：Clifford Group Equivariant Neural Networks。从 Clifford 代数中识别和研究 Clifford group 子群，形成了一个直接在向量基础上运算、高效推广到任意维数的 Equivariant Neural Network 层。从三维 N 体实验、四维 Lorentz-equivariant 高能物理实验到五维凸包实验都获得了最新的技术成果。

Clifford 群等变神经网络

Clifford Group Equivariant Neural Networks

Unitary $k$-designs are finite ensembles of unitary matrices that approximate
the Haar distribution over unitary matrices. Several ensembles are known to be
2-designs, including the uniform distribution over the Clifford group, but no
family of ensembles was previously known to form a 3-design. We prove that the
Clifford group is a 3-design, showing that it is a better approximation to
Haar-random unitaries than previously expected. Our proof strategy works for
any distribution of unitaries satisfying a property we call Pauli 2-mixing and
proceeds without the use of heavy mathematical machinery. We also show that the
Clifford group does not form a 4-design, thus characterizing how well random
Clifford elements approximate Haar-random unitaries. Additionally, we show that
the generalized Clifford group for qudits is not a 3-design unless the
dimension of the qudit is a power of 2.

本文证明了 Clifford 群是 3 设计，是以前所预期的逼近 Haar 随机酉矩阵更好的途径。我们的证明策略适用于任何满足我们所谓的 Pauli 2 混合属性的酉分布，且不需要使用沉重的数学工具。此外，我们还表明广义 Clifford 群不是 3 设计，除非 qudit 的维度是 2 的幂。