# Nyquist–Shannon sampling theorem for Quantum Evolution

In classical digital signal processing one can try to identify the dynamics of a system by sampling its evolution from an initial time $$t_0$$ to a final time $$t_1$$. Sampling $$N$$ times results in a discrete recreation of its evolution and one can try to reconstruct with various methods the actual continuous time evolution later.

By the Nyquist–Shannon sampling theorem if $$N$$ is sufficiently large then the discrete dynamics error should not be conceivable (e.g. if we are sampling the sound waves from a piano). Thus, to the observer non-observable.

Question

Is there some analog theorem or application of the Nyquist–Shannon sampling theorem when one wants to sample the evolution of a quantum state evolving under some Hamiltonian $$\hat H$$? (e.g. a qubit within some superconducting processor.)

One would record a time-series $$\{|\psi_0\rangle, \ldots,|\psi_N\rangle \}$$ where,

$$|\psi_i\rangle = e^{-i\hat H (t_i-t_0)}|\psi_0\rangle.$$

Could one prepare a state at least as many times as the state is supposed to be sample, say $$N$$ and allow evolution each time measuring with frequency $$1/N$$? I think the answer here is yes.

What can be said then about the state evolution and where could "quantum noise" ruin the reconstruction of the dynamics compared to a classical system?

Some errors would originate from the error reconstruction, but what other sources of error one would have to tackle?