The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limit

Abstract

In this paper, we study complex-valued neural network (CVNNs) with tensor-valued hidden-to-output weights within the framework of neural-network quantum field theory (NN-QFT). For standard CVNNs with scalar weights, we derive the generating functional and identify the exact Gaussian process that arises in the infinite-width limit, together with its associated effective quantum state. When the last-layer weights are promoted to Clifford-algebra-valued tensors, the network output becomes complex matrix-valued, and a fermion-like sign structure in the large-width correlation functions of the network output is induced. We show that, in the infinite-width limit, correlators with equal numbers of $f^{\dag}$ and $f$ obey fermionic Wick rules and can be written as determinants built from a scalar Euclidean kernel $S(x,y)=\langle f^{\dag}(x)f(y)\rangle$. This provides a sign-structured extension of NN-QFT at the level of Euclidean correlators and Feynman rules, even though a microscopic Grassmann path integral representation for the network parameters has not yet been constructed. Our analysis thus pushes the NN-QFT correspondence beyond purely bosonic Gaussian fields and suggests a possible route to encoding fermion-like symmetries in neural architectures for QFT correspondence.