An important question in quantum simulation is the certication of the quantumsimulators with proper readout. We examine how a detector's correlator changes when coupled to a quantum simulator using a diagrammatic technique. From the correlation functions calculated from the diagrammatic technique, we can determine whether or not reliable detection of the simulator's correlator can be achieved. When reliable detection is not possible due to detector back-action, we examine the situations when the back-action can be negligible. In particular, we study a cavity detector coupled to a Transverse Field Ising Model. We use a similar diagrammatic technique to study the interaction between a cavity and a qubit in the ultrastrong coupling regime. This cavity-qubit system is of importance in quantum computing and is a fundamental model in cavity QED. Ultrastrong coupling strength enables novel approaches for quantum logic operations. Our approach provides a fresh perspective on calculating the transmission spectra and the imacpt of the ultrastrongly coupled cavity on the qubit behavior.