Robustness and sensitivity to imperfections in the dynamics of general observables within near-term quantum simulators

We analyze the robustness of analog quantum simulators in the presence of weak perturbations. To this end, we focus on the dynamics of expectation values for generic observables and study the error arising from the imperfect operation of the device. We show that the properties of the error depend crucially on the observable considered as output of the simulator and we demostrate how even a low-fidelity quantum simulation is still able to reproduce the dynamics of other observables with a small relative error. To explain this, we define the observable purity, which characterizes the magnitude of the support of the observable in Hilbert space, and analytically show that these imperfect devices are able to reproduce the dynamics of low-purity observables (such as collective operators like the magnetization in a spin system) accurately, while the error in the expectation value of high-purity observables (such as projectors onto pure states) are much larger on average. We demonstrate our findings in a state-of-the-art 16-dimensional quantum simulator where, without assuming any particular error model, we observe this predicted behavior. This shows that these features are generic for a highly accurate device where we expect imperfections to be small.