McLachlan-projected reduced dynamics for ill-posed Schrödingerized backward diffusion

• URL: http://arxiv.org/abs/2605.17996v1
• Date: Mon, 18 May 2026 07:51:15 GMT
• Title: McLachlan-projected reduced dynamics for ill-posed Schrödingerized backward diffusion
• Abstract summary: Schrdingerization embeds the semidiscrete generator into Hermitian dynamics on an extended space.<n>We ask whether McLachlan projection onto a fixed low-dimensional frame supplies a structured regularizer.<n>We prove uniqueness of the reduced flow, Gram-norm conservation, a lifted--reduced gap bound in terms of that defect.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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. 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.

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