Quantum Injection Pathways for Implicit Graph Neural Networks

TL;DR

This paper introduces quantum injection pathways for graph DEQs, comparing three methods of coupling quantum signals, and demonstrates that independent injection achieves superior accuracy with fewer iterations on benchmark datasets.

Contribution

The paper formulates and compares three quantum signal injection methods for graph DEQs, providing contraction guarantees and empirical results showing the effectiveness of independent injection.

Findings

Independent injection achieves the best test accuracy.

Independent injection uses fewer forward-solver iterations.

Quantum injection pathways improve graph DEQ performance.

Abstract

Deep Equilibrium Models (DEQs) replace a stack of explicit layers with a single operator whose fixed point defines the output, giving the expressive power of an arbitrarily deep network at the memory cost of a single layer. Quantum Deep Equilibrium Models (QDEQs) bring this idea to quantum machine learning, offering an alternative to Parameterized Quantum Circuits (PQCs), whose depth is limited by hardware coherence and trainability. Here, we introduce, formulate, and compare three ways of coupling a quantum signal to graph DEQs, differing in where the signal enters the fixed-point operator. \textit{Independent} injection computes the quantum signal once per graph and forward fixed-point solve, and holds it fixed throughout the solve. \textit{State-dependent} injection instead recomputes the signal at every solver step and applies it to the current iterate. \textit{Backbone-dependent} injection likewise recomputes at every iteration but applies the signal to the classical backbone's output evaluated at the current iterate. We establish contraction guarantees for each variant under explicit assumptions on the Lipschitz constants of the classical backbone and the quantum signal. On the TU Dortmund graph-classification benchmarks NCI1, PROTEINS, and MUTAG, independent injection achieves the best test accuracy while using fewer forward-solver iterations than both the classical equilibrium baseline and the two dependent variants.