Quantum embedding tames the instability of running diffusion backward

Backward diffusion amplifies high-frequency noise exponentially, making direct numerical time-stepping unstable. This work uses Schrödingerization — lifting the problem into a larger Hermitian system — combined with McLachlan variational projection onto a fixed low-dimensional subspace to tame that instability. The paper proves uniqueness of the reduced flow, conservation of the Gram norm, and explicit error bounds tied to how well the subspace captures the full dynamics. A one-dimensional benchmark implemented in Qiskit Aer disentangles errors from the lifting step, the projection, and finite quantum-shot sampling separately.