McLachlan-projected reduced dynamics for ill-posed Schr\"odingerized backward diffusion

TL;DR

This paper introduces a McLachlan projection-based regularization method for the ill-posed backward diffusion problem, leveraging Schr"odingerization and reduced dynamics to improve stability and accuracy.

Contribution

It develops a structured regularization framework using McLachlan projection within Schr"odingerized dynamics, with theoretical guarantees and numerical validation.

Findings

Proves uniqueness and stability of the reduced flow.

Establishes a gap bound related to the projection defect.

Demonstrates improved numerical performance over classical filtering methods.

Abstract

Backward diffusion is a prototype ill-posed evolution: high spatial frequencies grow exponentially in time, so mesh-based time marching without explicit regularization is quickly overwhelmed by noise. Schr\"odingerization embeds the semidiscrete generator into Hermitian dynamics on an extended space; we ask whether McLachlan projection onto a fixed low-dimensional frame supplies a structured regularizer whose error budget can be read from a projection defect that separates full lifted propagation from the reduced trajectory. We prove uniqueness of the reduced flow, Gram-norm conservation, a lifted--reduced gap bound in terms of that defect, and perturbation estimates that highlight overlap-matrix conditioning when matrix elements are estimated statistically. We also spell out a fair classical baseline -- spectral low-pass or Tikhonov filtering on the same semidiscrete model, with bandwidth or ridge strength matched to the information content of the chosen frame -- so numerical contrasts isolate the Schr\"odingerized reduced pipeline rather than an unregularized Crank--Nicolson march that mainly showcases blow-up. A calibrated one-dimensional benchmark pairs a spectrally truncated reference with snapshot-built subspace evolution and finite-shot Qiskit Aer estimation, illustrating how lift, projection, and sampling layers contribute differently to the overall error.