Neural Controlled Differential Equations with Quantum Hidden Evolutions

TL;DR

The paper tackles improving sequential modeling by embedding quantum-inspired latent evolution into neural controlled differential equations (NCDEs). It models the hidden state as a complex wavefunction evolving via a Schrödinger-type dynamics, $dz_t = -i\hbar H z_t\,dX_t$, with unitary/orthogonal constraints enforced by ProjUNN or GeoTorch, and uses a collapse mapping $g$ to produce classification outputs $Y_t = \tilde{g}(z_t)$. Four variants are explored, differing in constraint method and concatenation order relative to the final linear layer, and are evaluated on a toy spiral dataset where they achieve 100% accuracy after training. The results suggest that complex-valued, unitary-evolving latent dynamics can be effectively integrated into NCDEs, offering stability and competitive performance, with future work aimed at deeper theoretical analysis and larger-scale benchmarks.

Abstract

We introduce a class of neural controlled differential equation inspired by quantum mechanics. Neural quantum controlled differential equations (NQDEs) model the dynamics by analogue of the Schrödinger equation. Specifically, the hidden state represents the wave function, and its collapse leads to an interpretation of the classification probability. We implement and compare the results of four variants of NQDEs on a toy spiral classification problem.