  Well-controlled quantum systems can potentially be used as quantum
simulators. However, a quantum simulator is inevitably perturbed by coupling to
additional degrees of freedom. This constitutes a major roadblock to useful
quantum simulations. So far there are only limited means to understand the
effect of perturbation on the results of quantum simulation. Here, we present a
method which, in certain circumstances, allows for the reconstruction of the
ideal result from measurements on a perturbed quantum simulator. We consider
extracting the value of the correlator $\langle\hat{O}^i(t)
\hat{O}^j(0)\rangle$ from the simulated system, where $\hat{O}^i$ are the
operators which couple the system to its environment. The ideal correlator can
be straightforwardly reconstructed by using statistical knowledge of the
environment, if any $n$-time correlator of operators $\hat{O}^i$ of the ideal
system can be written as products of two-time correlators. We give an approach
to verify the validity of this assumption experimentally by additional
measurements on the perturbed quantum simulator. The proposed method can allow
for reliable quantum simulations with systems subjected to environmental noise
without adding an overhead to the quantum system.
