Neural Group Actions

TL;DR

This paper introduces Neural Group Actions, a neural network framework for modeling symmetric transformations that satisfy finite group laws, extending involutive neural networks and demonstrating their application to quantum gate actions.

Contribution

It proposes a novel neural network architecture for modeling group actions, generalizing involutive neural networks and enforcing volume-preserving constraints.

Findings

Neural Group Actions can learn quantum gate operations satisfying group laws.

The framework generalizes involutive neural networks to arbitrary finite groups.

Experimental results show effective modeling of quaternion group actions on quantum states.

Abstract

We introduce an algorithm for designing Neural Group Actions, collections of deep neural network architectures which model symmetric transformations satisfying the laws of a given finite group. This generalizes involutive neural networks $\mathcal{N}$, which satisfy $\mathcal{N}(\mathcal{N}(x))=x$ for any data $x$, the group law of $\mathbb{Z}_2$. We show how to optionally enforce an additional constraint that the group action be volume-preserving. We conjecture, by analogy to a universality result for involutive neural networks, that generative models built from Neural Group Actions are universal approximators for collections of probabilistic transitions adhering to the group laws. We demonstrate experimentally that a Neural Group Action for the quaternion group $Q_8$ can learn how a set of nonuniversal quantum gates satisfying the $Q_8$ group laws act on single qubit quantum states.