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There have been several works on simulating ODE for classical systems like here and here. They are quantum techniques to solve the ODE related to classical systems.

A generic methodology is:

To solve the initial value problem $\frac{dy}{dt}=Ay+b$ for initial value $y(0)=y_0$, where $A$ can be non-Hermitian. Usually, the system is discretized to get a sparse linear system (SPS). This is solved using the Quantum linear system solver(QLSA). A caveat is that the final answer is encoded as Quantum state $|y(T)\rangle$.

The advantage is similar to one offered by QLSA (like HHL), i.e., $O(log(N))$ vs $N$ scaling with the linear system (SPS) size.

I do recognize that the advantage is due to QLSA, which requires sparse and low condition number requirements.

I am trying to internalize why the Quantum technique performs better than the classical technique for a classical system.

I do recognize (perhaps) my intuition for not expecting quantum speedup for a classical (non-Hermitian) problem is probably false.

I want a counter-argument for why the above fact is false in general.

(The reason for my intuition is the classical (non Hermitian) system has no quantum nature in it to be exploited by a quantum algorithm.)

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