Quasi-polar Decomposition of Quantum Neural Networks via Adaptive Non-local Observables

Abstract

We use Diagonal Adaptive Non-local Observables (DANO) as a canonical decomposition for studying Variational Quantum Circuit model evolution. Separating each learned observable into a diagonal spectrum and a unitary basis gives a quasi-polar description: the spectral weights are viewed as radial coordinates, while the unitary circuit serves as angular coordinates through Lie group identifications. This turns the training process into a trajectory in spectral and Lie-algebra space. Experiments on two classification tasks show that DANO radial spectral expansion correlates with accuracy. DANO angle coordinates reveal a dominant accuracy-correlated component. The framework provides a different perspective to characterize quantum model behavior.