McLachlan-projected reduced dynamics for ill-posed Schrödingerized backward diffusion
For researchers working on ill-posed inverse problems, this provides a structured regularization framework with theoretical guarantees, though the method is incremental as it combines existing techniques.
The paper addresses ill-posed backward diffusion by embedding it into Schrödinger dynamics and using McLachlan projection onto a low-dimensional frame as a regularizer. It proves uniqueness, conservation, and error bounds, and demonstrates the approach on a 1D benchmark with Qiskit Aer simulation.
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. Schrödingerization 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 Schrödingerized 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.