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I am reading about algorithms to simulate Hamiltonian evolution by means of quantum computers, e.g. a transverse field Ising model. As far as I see one is interested in getting expectation values of observables of interest. Generally, for an observable, $\mathcal{O}$, one computes the expectation value $\mathrm{Tr}(\mathcal{O}\rho)$.

What I was wondering is if the derivatives of such quantity over time, i.e. $\frac{d\mathrm{Tr}(\mathcal{O}\rho)}{dt}$ are also quantities of interest for some simulation problems. Also, are there any known methods to compute such derivatives?

Josu Etxezarreta Martinez
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You might need the Heisenberg picture of quantum dynamics, i.e., the state does not change with time, while observable $\hat{O}$ satisfies $$\frac{d \hat{O}}{dt} = \frac{i}{\hbar} [\hat{H}, \hat{O}] +\frac{\partial\hat{O}}{\partial t}$$ where $\hat{H}$ is the Hamiltonian. If the observable is time-independent $\frac{\partial\hat{O}}{\partial t}=0$, the time derivative of expectation value $$\frac{d \langle \hat{O}\rangle}{dt} = \frac{i}{\hbar} \langle [\hat{H}, \hat{O}]\rangle$$ can be calculated by the expectation of the commutator $[\hat{H},\hat{O}]$ at time $t$. Hope it helps. Correct me if it has mistake.

Jue
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